Integration as inverse operator

Derivative of \(\ln(x)\)

By Definition of the Derivative, we can write
\[f'(x) = \lim_{h \to 0} \frac{\ln(x+h) - \ln(x)}{h}\] Applying Logarithmic Properties, we get \[f'(x) = \lim_{h \to 0} \frac{1}{h} \ln\left(\frac{x+h}{x}\right) = \lim_{h \to 0} \frac{1}{h} \ln\left(1 + \frac{h}{x}\right)\] Using the Power Rule for Logs, we get \[f'(x) = \lim_{h \to 0} \ln\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\] Manipulation for the Constant \(e\), we get \[f'(x) = \lim_{h \to 0} \ln\left[\left(1 + \frac{h}{x}\right)^{\frac{x}{h}}\right]^{\frac{1}{x}}\] Evaluating the Limit, we get \[f'(x) = \frac{1}{x} \ln \left[ \lim_{u \to 0} (1+u)^{\frac{1}{u}} \right]\] So we have \[f'(x) = \frac{1}{x} \cdot 1 = \frac{1}{x}\]

Derivative of \(\sin(x)\)

By Definition of the Derivative, we get \[f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}\] Now we Apply Sum Identity, then \[f'(x) = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h}\] We Rearrange and Factor, then \[f'(x) = \lim_{h \to 0} \left[ \frac{\sin x (\cos h - 1)}{h} + \frac{\cos x \sin h}{h} \right]\] \[f'(x) = \sin x \cdot \lim_{h \to 0} \left( \frac{\cos h - 1}{h} \right) + \cos x \cdot \lim_{h \to 0} \left( \frac{\sin h}{h} \right)\] We Evaluate Special Limits, then \[\lim_{h \to 0} \frac{\sin h}{h} = 1 \quad \text{and} \quad \lim_{h \to 0} \frac{\cos h - 1}{h} = 0\] Now, we Substitute and Simplify, then \[f'(x) = \sin x \cdot (0) + \cos x \cdot (1)\] \[f'(x) = \cos x\]

Derivative of \(e^x\)

By Definition of the Derivative, we get \[f'(x) = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h}\] Using the exponent rule \(e^{x+h} = e^x \cdot e^h\), we get \[f'(x) = \lim_{h \to 0} \frac{e^x e^h - e^x}{h}\] \[f'(x) = \lim_{h \to 0} \frac{e^x(e^h - 1)}{h}\] By Extracting Constants from the Limit, we get \[f'(x) = e^x \cdot \lim_{h \to 0} \frac{e^h - 1}{h}\] Now we Evaluate the Limit, then \[\lim_{h \to 0} \frac{e^h - 1}{h} = 1\] Hence \[f'(x) = e^x \cdot 1 = e^x\]

Derivative of \(\sin^{-1}(x)\)

Initially we set up the inverse relationship, thus we suppose \(y = \sin^{-1}(x)\). This gives \[\sin(y) = x\] Now, we Differentiate implicitly with respect to \(x\), then \[\frac{d}{dx}[\sin(y)] = \frac{d}{dx}[x]\] Using the chain rule on the left side, we get \[\cos(y) \cdot \frac{dy}{dx} = 1\] Now, we solve for \(\frac{dy}{dx}\), then \[\frac{dy}{dx} = \frac{1}{\cos(y)}\] \[\frac{dy}{dx} = \frac{1}{\sqrt{1-sin ^2y}}\] Hence, we have \[\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}\]

Integration of \( \int \log (x) dx\)

We choose Integration by Parts, in which
\(u=\log (x)\) and \(v=1\)
Now , the integration is

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
+	\(\log (x)\)↘	\(1\)
-	\(\frac{1}{x}\)→	↘\(x\)	\(x \ln(x)\)
\(+\)			\(-\int \frac{1}{x} \cdot x \, dx\)

Hence, the integral is
\( \int \log (x) dx = x \log (x) - \int dx = x \log (x) - x + c\)

Integration of \( \int x^2 \sin x dx\)

We choose Integration by Parts, in which
\(u=x^2\) and \(dv=\sin(x)dx\)
Now, the integration is

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
+	\(x^2\)↘	\(\sin(x)\)
-	\(2x\)↘	\(-\cos(x)\)	\(-x^2 \cos(x)\)
+	\(2\)↘	\(-\sin(x)\)	\(+2x \sin(x)\)
-	\(0\)	\(\cos(x)\)	\(+2 \cos(x)\)

Hence, the integral is
\( \int x^2 \sin (x) dx = -x^2 \cos (x) + 2x \sin (x) + 2 \cos (x) + c \)

Integration of \( \int x e^x dx\)

We choose Integration by Parts, in which
\(u=x\) and \(dv=e^x dx\)
Now, the integration is     

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
	\(x\)↘	\(e^x\)
+	\(1\)↘	\(e^x\)	\(x e^x\)
-	\(0\)	\(e^x\)	\(-e^x\)

Hence, the integral is
\( \int x e^x dx = x e^x - e^x + c = e^x(x - 1) + c \)

Integration of \( \int e^x \sin x dx\)

We choose Integration by Parts, in which
\(u=\sin(x)\) and \(dv=e^x\)
Now, the integration is

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
+	\(\sin(x)\)↘	\(e^x\)
-	\(\cos(x)\)↘	\(e^x\)	\(e^x \sin(x)\)
+	\(-\sin(x)\) \(\rightarrow\)	\(e^x\)	\(-e^x \cos(x)\)

Hence, the integral is
\( \int e^x \sin (x) dx =e^x \sin(x) - e^x \cos(x) + \int (- \sin x) e^x dx \)
or\( 2\int e^x \sin (x) dx =e^x \sin(x) - e^x \cos(x) \)
or\( \int e^x \sin (x) dx =\frac{1}{2} [e^x \sin(x) - e^x \cos(x)] +c\)

Integration of \( \int e^{2x} \sin x dx\)

We choose Integration by Parts, in which
\(u=\sin(x)\) and \(dv=e^{2x}\)
Now, the integration is

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
	\(\sin(x)\)↘	\(e^{2x}\)
+	\(\cos(x)\)↘	\(\frac{1}{2}e^{2x}\)	\(\frac{1}{2}e^{2x} \sin(x)\)
-	\(-\sin(x)\) \(\rightarrow\)	\(\frac{1}{4}e^{2x}\)	\(-\frac{1}{4}e^{2x} \cos(x)\)

Hence, the integral is
\( \int e^{2x} \sin (x) dx =\frac{1}{2} e^{2x} \sin(x) - \frac{1}{4} e^{2x} \cos(x) - \int \frac{1}{4} e^{2x} \sin(x) dx \)
or\( \frac{5}{4} \int e^{2x}\sin (x) dx =\frac{1}{2} e^{2x} \sin(x) - \frac{1}{4} e^{2x} \cos(x) +c\)
or\( \int e^{2x}\sin (x) dx =\frac{1}{5} [2 e^{2x} \sin(x) - e^{2x} \cos(x)] +c\)

Integration of \( \int \sin^{-1} (x) dx\)

We choose Integration by Parts, in which
\(u=\sin^{-1} (x)\) and \(v=1\)
Now, the integration is     

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
	\(\sin^{-1} (x)\)↘	\(1\)
+	\(\frac{1}{\sqrt{1-x^2}}\)↘	\(x\)	\(x \sin^{-1}(x)\)
\(-\)		↘	\(-\int \frac{x}{\sqrt{1-x^2}} dx\)

Hence, the integral is
\( \int \sin^{-1} (x) dx = x \sin^{-1} (x) + \sqrt{1-x^2} + c \)

Integration of \( \int \log x \sin^{-1} x \, dx\)

We choose Integration by Parts, in which
\(u=\sin^{-1}(x)\) and \(dv=\ln(x)dx\)
Now, the integration is     

\(+/-\)	\(u\) (Derivative)	\(dv\) (Integral)	Product
	\(\sin^{-1}(x)\)↘	\(\ln(x)\)
+	\(\frac{1}{\sqrt{1-x^2}}\)↘	\(x\ln(x)-x\)	\((x\ln x - x)\sin^{-1}(x)\)
-		↘	\(-\int \frac{x\ln(x)-x}{\sqrt{1-x^2}} dx\)

Hence, the integral is
\( \int \ln(x) \sin^{-1}(x) dx = (x\ln x - x)\sin^{-1}(x) - \int \frac{x\ln x - x}{\sqrt{1-x^2}} dx + c \)

Integration of \( \int \tan x \, dx\)

We choose substitution, in which
\(u=\cos(x)\) then \(du=-\sin(x)dx\)
Now, the integration is
\( \int \tan(x) dx = \int \dfrac{\sin x}{\cos x} dx = -\int \dfrac{1}{u}du = -\log|\cos(x)| + c = \log|\sec(x)| + c \)

Integration of \( \int \cot x \, dx\)

We choose substitution, in which
\(u=\sin(x)\) then \(du=\cos(x)dx\)
Now, the integration is
\( \int \cot(x) dx =\int \dfrac{\cos x}{\sin x} dx =\int \dfrac{1}{u}du= \log|\sin(x)| + c \)

Integration of \( \int \sec x \, dx\)

We choose substitution, in which
\(u=\sec(x) + \tan(x)\) then \(du=(\sec x \tan x + \sec^2 x)dx\)
Now, the integration is
\( \int \sec(x) dx = \int \sec x \cdot \dfrac{\sec x + \tan x}{\sec x + \tan x} dx = \int \dfrac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} dx = \int \dfrac{1}{u}du = \log|\sec(x) + \tan(x)| + c \)

Integration of \( \int \csc x \, dx\)

We choose substitution, in which
\(u=\csc(x) + \cot(x)\) then \(du=(-\csc x \cot x - \csc^2 x)dx\)
Now, the integration is
\( \int \csc(x) dx = \int \csc x \cdot \dfrac{\csc x + \cot x}{\csc x + \cot x} dx = \int \dfrac{\csc^2 x + \csc x \cot x}{\csc x + \cot x} dx = -\int \dfrac{1}{u}du = -\log|\csc(x) + \cot(x)| + c \)

Integration of \( \int a^x \, dx\)

We choose substitution, in which
\(a^x = e^{x \ln a}\), let \(u = x \ln a\) then \(du = \ln a \, dx\)
Now, the integration is
\( \int a^x dx = \int e^{x \ln a} dx = \frac{1}{\ln a} \int e^u du = \frac{1}{\ln a} e^u + c = \dfrac{a^x}{\ln a} + c \)

Introduction

In calculus, differentiation and integration are two fundamental operations that are inversely related to each other. If you differentiate a function, you find its rate of change or slope at any given point. Integration, on the other hand, finds the accumulation or total value of a function over a given interval. When we say integration is the inverse operation of differentiation, we mean that integration "undoes" the effect of differentiation, and vice versa. More formally, if you integrate the derivative of a function, you'll get back the original function (up to a constant). Similarly, if you differentiate the integral of a function, you'll also get back the original function (up to a constant). Mathematically, this is represented by the Fundamental Theorem of Calculus

Formula for Integral

Integral of algebraic functions_1

SN	Derivative	Integral
1	\( \frac{d}{dx} \left ( x \right) =1 \)	\( \int \left ( 1 \right) dx=x+c \)
2	\( \frac{d}{dx} \left ( ax \right) =a \)	\( \int \left ( a \right) dx=ax+c \)
3	\( \frac{d}{dx} \left ( x^n \right) =n x^{n-1} \)	\( \int \left ( x^n \right) dx=\frac{x^{n+1}}{n+1}+c;n \ne -1 \)
4	\( \frac{d}{dx} (ax+b)^n=na (ax+b)^{n-1}\)	\( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}+c ;n \ne -1\)

Integral of algebraic functions_2

SN	Derivative	Integral
1	\( \frac{d}{dx} \log|x| =\frac{1}{x}\)	\( \int \frac{1}{x} dx=\log |x|+c \)
2	\( \frac{d}{dx} \log |ax+b| = \frac{a}{ax+b}\)	\( \int \frac{dx}{ax+b} dx= \frac{1}{a} \log |ax+b|+c\)

Integral of trigonometric functions

SN	Derivative	Integral
1	\( \frac{d}{dx} \left ( \cos x\right) =- \sin x\)	\( \int \left ( \sin x \right) dx=- \cos x+c \)
2	\( \frac{d}{dx} \left ( \sin x\right) = \cos x\)	\( \int \left ( \cos x \right) dx=- \sin x+c \)
3	\( \frac{d}{dx} \left ( \tan x\right) = \sec ^2x\)	\( \int \left ( \sec ^2x \right) dx= \tan x+c \)
		\( \int \left ( \tan x \right) dx=- \log |\cos x|+c \)
4	\( \frac{d}{dx} \left ( \cot x\right) = -\csc ^2x\)	\(\int \left ( \csc ^2x \right) dx= -\cot x+c \)
		\( \int \left ( \cot x \right) dx= \log |\sin x|+c \)
5	\( \frac{d}{dx} \left ( \sec x\right) = \sec x \tan x\)	\( \int \left ( \sec x \tan x \right) dx= \sec x+c \)
		\( \int \left ( \sec x \right) dx= \log |\sec x+\tan x|+c \)
6	\( \frac{d}{dx} \left ( \csc x\right) = -\csc x \cot x\)	\( \int \left ( \csc x \cot x \right) dx= -\csc x+c \)
		\( \int \left ( \csc x \right) dx= -\log |\csc x+\cot x|+c \)

Integral of exponential/logarithm functions

SN	Derivative	Integral
1	\( \frac{d}{dx} e^x = e^x\)	\( \int e^x dx=e^x+c\)
2	\( \frac{d}{dx} e^{ax} = ae^{ax}\)	\( \int e^{ax} dx=\frac{1}{a} e^{ax}+c\)
3	\( \frac{d}{dx} a^{x} = a^{x} \log a\)	\( \int a^{x} dx=\frac{a^{x}}{\log a} +c\)

Integral of inverse functions

SN	Derivative	Integral
1	\( \frac{d}{dx} \left ( \sin ^{-1} x\right) = \frac{1}{\sqrt{1-x^2}}\)	\(\int \left ( \frac{1}{\sqrt{1-x^2}} \right) dx=\sin ^{-1} x+c \)
2	\( \frac{d}{dx} \left ( \tan ^{-1} x\right) = \frac{1}{1+x^2}\)	\( \int \left ( \frac{1}{1+x^2} \right) dx=\tan ^{-1} x+c \)
3	\( \frac{d}{dx} \left ( \sec ^{-1} x\right) = \frac{1}{|x| \sqrt{x^2-1}} \)	\( \int \left ( \frac{1}{|x| \sqrt{x^2-1}} \right) dx=\sec ^{-1} x+c \)
4	\( \frac{d}{dx} \left ( \cos ^{-1} x\right) = \frac{-1}{\sqrt{1-x^2}}\)	\(\int \left ( \frac{-1}{\sqrt{1-x^2}} \right)dx=\cos ^{-1} x+c \)
5	\( \frac{d}{dx} \left ( \cot ^{-1} x\right) = \frac{-1}{1+x^2}\)	\( \int \left ( \frac{-1}{1+x^2} \right) dx=\cot ^{-1} x+c \)
6	\( \frac{d}{dx} \left ( \csc ^{-1} x\right) = \frac{-1}{|x| \sqrt{x^2-1}} \)	\( \int \left ( \frac{-1}{|x| \sqrt{x^2-1}} \right) dx=\csc ^{-1} x+c \)

Basic Integral of algebraic functions: Activity 1

Compute the following integrals.

1. \( \int 60 dt\)

   Solution 👉 Click Here

   Given integral is
   \( \int 60 dt \)
   or\( 60 \int dt\)
   or\( 60 (t)+c\) \(\int x dx=x\)
   or\( 60 t +c\)
2. \( \int 2x dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int 2x dx\)
   or\( 2 \int x dx\)
   or\( 2 \left ( \frac{x^2}{2} \right )+c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}+c\)
   or\( x^2+c\)
3. \( \int 5x^3 dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int 5x^3 dx\)
   or\( 5 \int x^3 dx\)
   or\( 5 \frac{x^4}{4}+c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
   or\( \frac{5}{4} x^4 +c\)
4. \( \int 7x^{5/2}dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int 7x^{5/2}dx\)
   or\(7 \int x^{5/2}dx \)
   or\( 7 \frac{x^{7/2}}{7/2}+c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
   or\( 2 x^{7/2} +c\)
5. \( \int 4x^{-5} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int 4x^{-5} dx\)
   or\(4 \int x^{-5} dx \)
   or\( 4 \frac{x^{-4}}{-4} +c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
   or\( - x^{-4} +c\)
6. \( \int 2x^{-\frac{7}{2}} dx \)

   Solution 👉 Click Here

   Given integral is
   \( \int 2x^{-\frac{7}{2}} dx \)
   or\( 2 \int x^{-\frac{7}{2}} dx \)
   or\( 2 \frac{x^{-\frac{5}{2}}}{-\frac{5}{2}} +c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
   or\( -\frac{4}{5} x^{-\frac{5}{2}} +c\)
7. \( \int (2x+4)dx \)

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   Given integral is
   \( (2x+4)dx \)
   or\( \int 2x dx + \int 4 dx\)
   or\( 2 \int x dx + 4 \int dx\)
   or\( 2 \left ( \frac{x^2}{2} \right )+4 (x) +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x\)
   or\( x^2+x+c\)
8. \( \int (w^2-3)dw \)

   Solution 👉 Click Here

   Given integral is
   \( \int (w^2-3)dw \)
   or\( \int w^2 dw - \int 3 dw\)
   or\( \int w^2 dw - 3 \int dw\)
   or\( \left ( \frac{w^3}{3} \right )-3 (w) +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x\)
   or\( \frac{w^3}{3} -3w+c\)
9. \( \int (x^2+2) dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int (x^2+2) dx \)
   or\( \int x^2 dx + \int 2 dx\)
   or\( \int x^2 dx + 2 \int dx\)
   or\( \left ( \frac{x^3}{3} \right )+2 (x) +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x\)
   or\( \frac{x^3}{3}+2x+c\)
10. \( \int (3x^2+2x+1) dx\)

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    Given integral is
    \( \int (3x^2+2x+1) dx \)
    or\( \int 3 x^2 dx + \int 2x dx+\int 1 dx\)
    or\( 3 \int x^2 dx + 2 \int x dx+\int dx\)
    or\( 3 \left ( \frac{x^3}{3} \right )+2 \left ( \frac{x^2}{2} \right ) +x+c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x\)
    or\( x^3+x^2+x+c\)
11. \( \int (2x+1)(3x+2) dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (2x+1)(3x+2) dx\)
    or\( \int (6x^2+7x+2) dx\)
    or\( \int 6 x^2 dx + \int 7x dx+\int 2 dx\)
    or\( 6 \int x^2 dx + 7 \int x dx+2 \int dx\)
    or\( 6 \left ( \frac{x^3}{3} \right )+7 \left ( \frac{x^2}{2} \right ) +2x+c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x\)
    or\( 2x^3+\frac{7}{2}x^2+2x+c\)
12. \( \int (4x^{1/3}+5x^{2/3}+\frac{1}{x^2}) dx \)

    Solution 👉 Click Here

    Given integral is
    \( \int (4x^{1/3}+5x^{2/3}+\frac{1}{x^2}) dx \)
    or\( \int 4x^{1/3} dx+\int 5x^{2/3} dx+\int \frac{1}{x^2} dx \)
    or\( 4 \int x^{1/3} dx+5\int x^{2/3} dx+\int x^{-2} dx \)
    or\( 4 \left ( \frac{x^{4/3}}{4/3} \right )+5 \left ( \frac{x^{5/3}}{5/3} \right )+\frac{x^{-1}}{-1} +c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
    or\( 3 x^{4/3}+3 x^{5/3}-\frac{1}{x} +c\)
13. \( \int (x^{3/4}+x^{1/2}+4x^{1/3} )dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (x^{3/4}+x^{1/2}+4x^{1/3} )dx \)
    or\( \int x^{3/4}dx + \int x^{1/2}dx+ \int 4 x^{1/3} dx \)
    or\( \int x^{3/4}dx + \int x^{1/2}dx+ 4 \int x^{1/3} dx \)
    or\( \left ( \frac{x^{7/4}}{7/4} \right )+ \left ( \frac{x^{3/2}}{3/2} \right )+4 \left ( \frac{x^{4/3}}{4/3} \right ) +c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
    or\( \frac{4}{7}x^{7/4}+ \frac{2}{3}x^{3/2}+3x^{5/3} +c\)
14. \( \int (x^2-\frac{1}{x^2}) dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (x^2-\frac{1}{x^2}) dx \)
    or\( \int (x^2-x^{-2}) dx \)
    or\( \int x^2 dx + \int x^{-2} dx \)
    or\( \left ( \frac{x^3}{3} \right )+ \left ( \frac{x^{-1}}{-1} \right )+c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
    or\( \frac{x^3}{3} -x^{-1}+c\)
    or\( \frac{x^3}{3} -\frac{1}{x}+c\)
    
15. \( \int \sqrt{x} (x^2-5) dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \sqrt{x} (x^2-5) dx \)
    or\( \int x^{1/2} (x^2-5) dx \)
    or\( \int (x^{5/2} -5 x^{1/2}) dx \)
    or\( \int x^{5/2} dx -5 \int x^{1/2} dx \)
    or\( \left ( \frac{x^{7/2}}{7/2} \right )-5 \left ( \frac{x^{3/2}}{3/2} \right )+c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
    or\( \frac{2}{7} x^{7/2} -\frac{10}{3} x^{3/2}+c\)
16. \( \int (x^2+3x+5)x^{-1/2}dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (x^2+3x+5)x^{-1/2}dx \)
    or\( \int \left ( x^{3/2}+3 x^{1/2}+5x^{-1/2} \right ) dx \)
    or\( \int x^{3/2} dx+ \int 3 x^{1/2} dx +\int 5x^{-1/2} dx \)
    or\( \int x^{3/2} dx+ 3 \int x^{1/2} dx + 5 \int x^{-1/2} dx \)
    or\( \frac{x^{5/2}}{5/2}+3 \left ( \frac{x^{3/2}}{3/2} \right ) +5 \left ( \frac{x^{1/2}}{1/2} \right ) +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
    or\( \frac{2}{5} x^{5/2} +2 x^{3/2}+10 x^{1/2}+c\)
17. \( \int \left ( \sqrt{x}-\frac{1}{\sqrt{x}} \right ) dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \left ( \sqrt{x}-\frac{1}{\sqrt{x}} \right ) dx\)
    or\( \int \left ( x^{1/2}-x^{-1/2} \right ) dx \)
    or\( \int x^{1/2} dx- \int x^{-1/2} dx \)
    or\( \frac{x^{3/2}}{3/2}-\frac{x^{1/2}}{1/2} +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
    or\( \frac{2}{3}x^{3/2} - 2 x^{1/2}+c \)
18. \( \int (x-3)^2 dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (x-3)^2 dx \)
    or\( \frac{(x-3)^3}{3 } +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
19. \( \int (4x+5)^4dx \)

    Solution 👉 Click Here

    Given integral is
    \( \int (4x+5)^4 dx \)
    or\( \frac{(4x+5)^5}{4 \times 5 } +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{(4x+5)^5}{20 } +c \)
20. \( \int (3x+5)^4 dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (3x+5)^4 dx \)
    or\( \frac{ (3x+5)^5}{3 \times 5 } +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{(3x+5)^5}{15 } +c \)
21. \( \int (a-bx)^5 dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (a-bx)^5 dx \)
    or\( \frac{ (a-bx)^6}{b \times 6 } +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{(a-bx)^6}{6b } +c \)
22. \( \int (c+dx)^{-3/2}dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (c+dx)^{-3/2}dx \)
    or\( \frac{ (c+dx)^{-1/2}}{d \times (-1/2) } +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( -2 (c+dx)^{-1/2} +c \)
    or\( -\frac{2}{\sqrt{c+dx} } +c \)
    
23. \( \int \frac{1}{\sqrt{2x+7} }dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \frac{1}{\sqrt{2x+7} }dx \)
    or\( \int (2x+7)^{-1/2} dx \)
    or\( \frac{(2x+7)^{1/2}}{(1/2) \times 2} dx \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \sqrt{2x+7} +c \)
24. \( \int \left ( x+\frac{1}{(x+3)^2} \right ) dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \left ( x+\frac{1}{(x+3)^2} \right ) dx \)
    or\( \int x dx+ \int \frac{1}{(x+3)^2} dx \)
    or\( \int x dx+ \int (x+3)^{-2} dx \)
    or\( \frac{x^2}{2}+ \frac{(x+3)^{-1}}{-1} +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1},\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{x^2}{2} - \frac{1}{(x+3)}+c \)
25. \( \int \left ( 4+\frac{1}{(5x+1)^2} \right ) dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \left ( 4+\frac{1}{(5x+1)^2} \right ) dx \)
    or\( \int 4 dx+ \int \frac{1}{(5x+1)^2} dx \)
    or\( 4 \int dx+ \int (5x+1)^{-2} dx \)
    or\( 4x + \frac{(5x+1)^{-1}}{-1 \times 5} +c \) \(\int x^n dx=\frac{x^{n+1}}{n+1},\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( 4x - \frac{1}{5(5x+1)}+c \)

Basic Integral of algebraic functions: Activity 2

Compute the following integrals.

1. \( \int \frac{1}{\sqrt{x+1}-\sqrt{x}}dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{1}{\sqrt{x+1}-\sqrt{x}}dx \)
   or\( \int \frac{1}{\sqrt{x+1}-\sqrt{x}} \times \frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}} dx \)
   or\( \int \frac{\sqrt{x+1}+\sqrt{x}}{(x+1)-x}dx \)
   or\( \int \left ( \sqrt{x+1}+\sqrt{x} \right ) dx \)
   or\( \int \left ( (x+1)^{1/2}+x^{1/2} \right ) dx \)
   or\( \frac{(x+1)^{3/2}}{3/2}+\frac{x^{3/2}}{3/2} +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{2}{3}(x+1)^{3/2}+\frac{2}{3}x^{3/2}+c \)
2. \( \int \frac{1}{\sqrt{x+a}-\sqrt{x-b}} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{1}{\sqrt{x+a}-\sqrt{x-b}} dx \)
   or\( \int \frac{1}{\sqrt{x+a}-\sqrt{x-b}} \times \frac{\sqrt{x+a}+\sqrt{x-b}}{\sqrt{x+a}+\sqrt{x-b}} dx \)
   or\( \int \frac{\sqrt{x+a}+\sqrt{x-b}}{(x+a)-(x-b)} dx \)
   or\( \int \frac{\sqrt{x+a}+\sqrt{x-b}}{(a+b)} dx \)
   or\( \frac{1}{(a+b)} \int \left [ (x+a)^{1/2}+(x-b)^{1/2} \right ] dx \)
   or\( \frac{1}{(a+b)} \left [ \frac{(x+a)^{3/2}}{3/2} +\frac{(x-b)^{3/2}}{3/2} \right ] +c \) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{1}{(a+b)} \left [\frac{2}{3} (x+a)^{3/2} +\frac{2}{3} (x-b)^{3/2} \right ] +c \)
3. \( \int \frac{1}{\sqrt{2x+1}-\sqrt{2x-3}}dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{1}{\sqrt{2x+1}-\sqrt{2x-3}}dx \)
   or\( \int \frac{1}{\sqrt{2x+1}-\sqrt{2x-3}} \times \frac{\sqrt{2x+1}+\sqrt{2x-3}}{\sqrt{2x+1}+\sqrt{2x-3}} dx \)
   or\( \int \frac{\sqrt{2x+1}+\sqrt{2x-3}}{(2x+1)-(2x-3)}dx \)
   or\( \int \frac{\sqrt{2x+1}+\sqrt{2x-3}}{4}dx \)
   or\(\frac{1}{4} \int \left [ (2x+1)^{1/2}+(2x-3)^{1/2} \right ] dx \)
   or\(\frac{1}{4} \left [ \frac{(2x+1)^{3/2}}{(3/2) \times 2 } +\frac{(2x-3)^{3/2}}{(3/2) \times 2 } \right ] +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{1}{12} \left [ (2x+1)^{3/2} +(2x-3)^{3/2} \right ] +c \)
4. \( \int \frac{1}{\sqrt{x+3}-\sqrt{x-1}} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{1}{\sqrt{x+3}-\sqrt{x-1}} dx \)
   or\( \int \frac{1}{\sqrt{x+3}-\sqrt{x-1}} \times \frac{\sqrt{x+3}+\sqrt{x-1}}{\sqrt{x+3}+\sqrt{x-1}}dx \)
   or\( \int \frac{\sqrt{x+3}+\sqrt{x-1}}{(x+3)-(x-1)}dx \)
   or\( \int \frac{\sqrt{x+3}+\sqrt{x-1}}{4}dx \)
   or\(\frac{1}{4} \int \left [ (x+3)^{1/2}+(x-1)^{1/2} \right ] dx \)
   or\(\frac{1}{4} \left [ \frac{(x+3)^{3/2}}{(3/2) } +\frac{(x-1)^{3/2}}{(3/2) } \right ] +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{1}{6} \left [ (x+3)^{3/2} +(x-1)^{3/2} \right ] +c \)
5. \( \int (2x+3)(4x+5)^4 dx \)

   Solution 👉 Click Here

   Given integral is
   \( \int (2x+3)(4x+5)^4 dx \)
   or\( \frac{1}{2} \int \left ( 4x+ 6 \right )(4x+5)^4 dx \)
   or\( \frac{1}{2} \int \left [ (4x+ 5)+1 \right ](4x+5)^4 dx \)
   or\( \frac{1}{2} \int \left [ (4x+5)^5+(4x+5)^4 \right ] dx \)
   or\( \frac{1}{2} \left [ \int (4x+5)^5 dx +\int (4x+5)^4 dx \right ] \)
   or\( \frac{1}{2} \left [ \frac{(4x+5)^6}{4.6} +\frac{(4x+5)^5}{4.5} \right ] +c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{1}{8} \left [ \frac{(4x+5)^6}{6} +\frac{(4x+5)^5}{5} \right ] +c \)
6. \( \int (3x+2)(2x+5)^3 dx \)

   Solution 👉 Click Here

   Given integral is
   \( \int (3x+2)(2x+5)^3 dx \)
   or\( \int 3 (x+\frac{2}{3})(2x+5)^3 dx \)
   or\( \int \frac{3}{2} (2x+\frac{4}{3})(2x+5)^3 dx \)
   or\( \frac{3}{2} \int [(2x+ 5) +(\frac{4}{3}-5)](2x+5)^3 dx \)
   or\( \frac{3}{2} \int \left [ (2x+5)^4 -\frac{11}{3} (2x+5)^3 \right ] dx\)
   or\( \frac{3}{2} \left [ \int (2x+5)^4 dx -\frac{11}{3} \int (2x+5)^3 dx \right ] \)
   or\( \frac{3}{2} \left [ \frac{(2x+5)^5}{5 \times 2} -\frac{11}{3} \frac{ (2x+5)^4}{4 \times 2} \right ]+c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \left [ \frac{3}{20} (2x+5)^5 -\frac{11}{60} (2x+5)^4 \right ]+c \)
   
7. \( \int x \sqrt{x+1} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int x \sqrt{x+1} dx \)
   or\( \int (x+1-1) \sqrt{x+1} dx \)
   or\( \int [(x+1)-1] (x+1)^{1/2} dx \)
   or\( \int \left [ (x+1)^{3/2}-(x+1)^{1/2} \right ] dx \)
   or\( \int (x+1)^{3/2} dx - \int (x+1)^{1/2} dx \)
   or\( \frac{(x+1)^{5/2}}{5/2} - \frac{(x+1)^{3/2}}{3/2} +c \) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   
8. \( \int 2x \sqrt{2x+3} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int 2x \sqrt{2x+3} dx \)
   or\( \int (2x+3-3) \sqrt{2x+3} dx \)
   or\( \int [(2x+3)-3] (2x+3)^{1/2} dx \)
   or\( \int \left [ (2x+3)^{3/2}-3 (2x+3)^{1/2} \right ] dx \)
   or\( \int (2x+3)^{3/2} dx - 3 \int (2x+3)^{1/2} dx \)
   or\( \frac{(2x+3)^{5/2}}{(5/2) \times 2 } - 3 \frac{(2x+3)^{3/2}}{(5/2) \times 2} +c \) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{(2x+3)^{5/2}}{5} - 3 \frac{(2x+3)^{3/2}}{5} +c \)
9. \( \int x \sqrt{ax+b}dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int x \sqrt{ax+b}dx \)
   or\( \int \frac{1}{a}(ax) \sqrt{ax+b}dx \)
   or\( \int \frac{1}{a}[(ax+b)-b] \sqrt{ax+b}dx\)
   or\( \int \frac{1}{a}[(ax+b)-b] (ax+b)^{1/2} dx \)
   or\( \frac{1}{a} \int \left [ (ax+b)^{3/2}-b(ax+b)^{1/2} \right ] dx \)
   or\( \frac{1}{a} \left [ \frac{(ax+b)^{5/2}}{\frac{5}{2} \times a} -b \frac{(ax+b)^{3/2}}{\frac{3}{2} \times a} \right ] +c\) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{1}{a^2} \left [ \frac{2}{5} (ax+b)^{5/2}-\frac{2b}{3} (ax+b)^{3/2}\right ]+c \)
10. \( \int 5x \sqrt{5x+2}dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int 5x \sqrt{5x+2}dx \)
    or\( \int [(5x+2)-2] \sqrt{5x+2}dx \)
    or\( \int [(5x+2)-2] (5x+2)^{1/2} dx \)
    or\( \int \left [ (5x+2)^{3/2}-2 (5x+2)^{1/2} \right ] dx \)
    or\( \left [ \frac{(5x+2)^{5/2}}{\frac{5}{2} \times 5} -2 \frac{(5x+2)^{3/2}}{\frac{3}{2} \times 5} \right ] +c\) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{2}{25} (5x+2)^{5/2} -\frac{4}{15} (5x+2)^{3/2}+c \)
11. \( \int (x+2) \sqrt {3x+2} dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (x+2) \sqrt {3x+2} dx \)
    or\( \frac{1}{3} \int (3x+6) \sqrt {3x+2} dx \)
    or\( \frac{1}{3} \int \left [ (3x+2)+4 \right ] (3x+2)^{1/2} dx\)
    or\( \frac{1}{3} \int \left [ (3x+2)^{3/2}+4 (3x+2)^{1/2} \right ] dx\)
    or\( \frac{1}{3} \left [ \frac{(3x+2)^{5/2}}{(5/2) \times 3} +4 \frac{(3x+2)^{3/2}}{(3/2) \times 3} \right ] +c\) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{2}{5} (3x+2)^{5/2} + \frac{8}{3} (3x+2)^{3/2} +c\)
12. \( \int (5x+3) \sqrt{4x+1} dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (5x+3) \sqrt{4x+1} dx \)
    or\( \frac{5}{4} \int (4x+\frac{12}{5}) \sqrt {4x+1} dx \)
    or\( \frac{5}{4} \int \left [ (4x+1) +(\frac{12}{5}-1) \right ] (4x+1)^{1/2} dx \)
    or\( \frac{5}{4} \int \left [ (4x+1)^{3/2}+ \frac{7}{5} (4x+1)^{1/2} \right ] dx\)
    or\( \frac{5}{4} \left [ \frac{(4x+1)^{5/2}}{(5/2) \times 4} +\frac{7}{5} \frac{(4x+1)^{3/2}}{(3/2) \times 4} \right ] +c\) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{1}{8} (4x+1)^{5/2} +\frac{7}{24} (4x+1)^{3/2}+c\)
13. \( \int (2x+3) \sqrt{3x+1}dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int (2x+3) \sqrt{3x+1}dx \)
    or\( \int 2 (x+\frac{3}{2}) \sqrt{3x+1}dx \)
    or\( \int \frac{2}{3} (3x+\frac{9}{2}) \sqrt{3x+1}dx \)
    or\( \frac{2}{3} \int [(3x+1)+(\frac{9}{2}-1)] \sqrt{3x+1}dx \)
    or\( \frac{2}{3} \int [(3x+1)+\frac{7}{2}] (3x+1)^{1/2} dx \)
    or\( \frac{2}{3} \int \left [ (3x+1)^{3/2}+\frac{7}{2} (3x+1)^{1/2} \right ] dx \)
    or\( \frac{2}{3} \left [ \frac{(3x+1)^{5/2}}{\frac{5}{2} \times 3} +\frac{7}{2} \frac{(3x+1)^{3/2}}{\frac{3}{2} \times 3} \right ] +c\) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{4}{45} (3x+1)^{5/2} +\frac{14}{27} (3x+1)^{3/2}+c \)
14. \( \int \frac{3x+4}{\sqrt{x+1}} dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \frac{3x+4}{\sqrt{x+1}} dx \)
    or\( \int 3 \frac{x+\frac{4}{3}}{\sqrt{x+1}} dx \)
    or\( 3 \int \frac{(x+1) +(\frac{4}{3}-1) }{\sqrt{x+1}} dx \)
    or\( 3 \int \frac{(x+1)}{\sqrt{x+1}} +\frac{\frac{1}{3}}{\sqrt{x+1}} dx \)
    or\( 3 \int (x+1)^{1/2} dx + \int (x+1)^{-1/2} dx \)
    or\( 3 \frac{(x+1)^{3/2}}{3/2} + \frac{(x+1)^{1/2}}{1/2}+c \) \(\int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( 2 (x+1)^{3/2} + 2 (x+1)^{1/2}+c \)
15. \( \int \frac{x+2}{\sqrt{x+1}} dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \frac{x+2}{\sqrt{x+1}} dx \)
    or\( \int \frac{(x+1)+1}{\sqrt{x+1}} dx \)
    or\( \int \left [ \frac{(x+1)}{\sqrt{x+1}}+\frac{1}{\sqrt{x+1}} \right ] dx \)
    or\( \int \left [ \sqrt{x+1}+\frac{1}{\sqrt{x+1}} \right ] dx \)
    or\( \int \left [ (x+1)^{1/2}+(x+1)^{-1/2} \right ] dx\)
    or\( \left [ \frac{(x+1)^{3/2}}{\frac{3}{2}} +\frac{(x+1)^{1/2}}{\frac{1}{2} } \right ] +c\) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{2}{3} (x+1)^{3/2} +\frac{1}{2} (x+1)^{1/2}+c \)
16. \( \int \frac{2x+1}{\sqrt{3x+2}}dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \frac{2x+1}{\sqrt{3x+2}}dx \)
    or\( \int 2 \frac{x+\frac{1}{2}}{\sqrt{3x+2}}dx \)
    or\( \int \frac{2}{3} \frac{3x+\frac{3}{2}}{\sqrt{3x+2}}dx \)
    or\( \int \frac{2}{3} \frac{[(3x+2)+(\frac{3}{2}-2)}{\sqrt{3x+2}}dx \)
    or\( \int \frac{2}{3} \frac{[(3x+2)-\frac{1}{2}}{\sqrt{3x+2}}dx \)
    or\( \frac{2}{3} \int \left [ \frac{(3x+2)}{\sqrt{3x+2}} -\frac{1}{2}\frac{1} {\sqrt{3x+2}} \right ] dx \)
    or\( \frac{2}{3} \int \left [ \sqrt{3x+2} -\frac{1}{2} \frac{1} {\sqrt{3x+2}} \right ]dx \)
    or\( \frac{2}{3} \int \left [ (3x+2)^{1/2} -\frac{1}{2} (3x+2)^{-1/2} \right ]dx \)
    or\( \frac{2}{3} \left [ \frac{(3x+2)^{3/2}}{\frac{3}{2} \times 3} -\frac{1}{2} \frac{(3x+2)^{1/2}}{\frac{1}{2} \times 3} \right ] +c\) \( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
    or\( \frac{4}{27} (3x+2)^{3/2} -\frac{2}{9} (3x+2)^{1/2}+c \)

Basic Integral of algebraic functions: Activity 3

Compute the following integrals.

1. \( \int \frac{(3x+2)}{(5x+1)^2} dx \)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{(3x+2)}{(5x+1)^2} dx \)
   or\( \int \frac{3(x+\frac{2}{3})}{(5x+1)^2} dx\)
   or\( \int \frac{\frac{3}{5}(5x+\frac{10}{3})}{(5x+1)^2} dx\)
   or\( \frac{3}{5} \int \frac{(5x+1)+[\frac{10}{3}-1]}{(5x+1)^2} dx\)
   or\( \frac{3}{5} \int \frac{(5x+1)+(\frac{7}{3})}{(5x+1)^2} dx\)
   or\( \frac{3}{5} \int \left [ \frac{(5x+1)}{(5x+1)^2}+\frac{7}{3} \frac{1}{(5x+1)^2} \right ] dx\)
   or\( \frac{3}{5} \int \left [ \frac{1}{5x+1}+\frac{7}{3} \frac{1}{(5x+1)^2} \right ] dx\)
   or\( \frac{3}{5} \left [ \frac{\log (5x+1)}{5}+\frac{7}{3} \frac{(5x+1)^3}{3 \times 5} \right ]+c\) \(\int \frac{dx}{ax+b}=\frac{\log(ax+b)}{a}, \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}\)
   or\( \frac{3}{5} \left [ \frac{\log (5x+1)}{5}+\frac{7}{45} (5x+1)^3 \right ]+c\)
2. \( \int \frac{x+3}{x-3} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{x+3}{x-3} dx \)
   or\( \int \frac{x-3+6}{x-3} dx\)
   or\( \int \left [\frac{x-3}{x-3}+\frac{6}{x-3} \right ] dx\)
   or\( \int \left [1+\frac{6}{x-3} \right ] dx\)
   or\( \left [\int 1 dx + \int \frac{6}{x-3} dx \right ] \)
   or\( \left [x + 6 \log (x-3) \right ] +c \) \(\int \frac{dx}{ax+b}=\frac{\log(ax+b)}{a}\)
3. \( \int \frac{3x^2-5x+2}{x}dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{3x^2-5x+2}{x}dx \)
   or\( \int \left ( \frac{3x^2}{x}-\frac{5x}{x}+\frac{2}{x} \right ) dx \)
   or\( \int 3x dx-\int 5 dx+\int \frac{2}{x} dx \)
   or\( 3 \int x dx-5 \int dx+\int \frac{2}{x} dx \)
   or\( 3 \left ( \frac{x^2}{2} \right )-5(x) + 2 \log x +c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x, \int \frac{1}{x}=\log x\)
   or\( \frac{3}{2}x^2-5x+2 \log x +c \)
4. \( \int \frac{x^2+3x+3}{x+2} dx \)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{x^2+3x+3}{x+2} dx \)
   or\( \int \frac{(x^2+3x+2)+1}{x+2} dx\)
   or\( \int \frac{(x+2)(x+1)+1}{x+2} dx \)
   or\( \int (x+1) + \frac{1}{x+2} dx \)
   or\( \frac{x^2}{2}+x+ \log (x+2)+c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x, \int \frac{1}{x}=\log x\)
5. \( \int \frac{ax^2+bx+c}{x^2} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{ax^2+bx+c}{x^2} dx \)
   or\( \int \left ( a+b \frac{1}{x} +c \frac{1}{x^2} \right ) dx\)
   or\( \int \left ( a+b \frac{1}{x} +c x^{-2} \right ) dx\)
   or\( a x +b \log x +c \frac{x^{-1}}{-1} +k\) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x, \int \frac{1}{x}=\log x\)
   or\( a x +b \log x -c \frac{1}{x} +k\)
6. \( \int \frac{3x-1}{x-2} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{3x-1}{x-2} dx\)
   or\( 3 \int \frac{x-\frac{1}{2}}{x-2} dx \)
   or\( 3 \int \frac{ (x-2)+(2-\frac{1}{2})}{x-2} dx \)
   or\( 3 \int \left [1+ \frac{ \frac{3}{2} }{x-2} \right ] dx \)
   or\( 3 \int \left [1+ \frac{3}{2} \frac{1 }{x-2} \right ] dx \)
   or\( 3 \left [x + \frac{3}{2} \log (x-2) \right ] +c \) \(\int dx=x, \int \frac{dx}{ax+b}=\frac{\log(ax+b)}{a}\)
7. \( \int \frac{x^2+5}{x+2} dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \frac{x^2+5}{x+2} dx \)
   or\( \int \frac{(x^2-4)+(5+4)}{x+2} dx \)
   or\( \int \frac{(x^2-4)+9}{x+2} dx\)
   or\( \int \left [ \frac{(x^2-4)}{x+2}+\frac{9}{x+2} \right ] dx\)
   or\( \int \left [ (x-2)+\frac{9}{x+2} \right ] dx\)
   or\( \left [ \frac{x^2}{2}-2x+ 9 \log (x+2) \right ] +c\) \(\int x^n dx=\frac{x^{n+1}}{n+1}, \int dx=x, \int \frac{dx}{ax+b}=\frac{\log(ax+b)}{a}\)

Basic Integral of trigonometric functions: Activity 4

Compute the following integrals.

1. \( \int \sin 5x dx\)

   Solution 👉 Click Here

   Given integral is
   \( \int \sin 5x dx \)
   or\( - \frac{\cos 5x}{5} +c\) \(\int \sin ax dx=\frac{- \cos ax}{a}\)
2. \( \int \sin^2 ax dx\)
   
3. \( \int \tan ^2 a x dx\)
   
4. \( \int \cos^4 n x dx\)
   
5. \( \int \cos (a^2x+b)dx\)
   
6. \( \int \frac{1}{\sec^2 xx \tan^2 x } dx\)
   
7. \( \int \sqrt{1+\cos n x} dx\)
   
8. \( \int \sqrt{1+\sin 2a x} dx\)
   
9. \( \int \frac{1}{1-\cos nx } dx\)
   
10. \( \int \sin (ax+b) dx\)
    
11. \( \int \sec^2 (2x+3) dx\)
    
12. \( \int \cos ^2 bx dx\)
    
13. \( \int \sin ^4 x dx\)
    
14. \( \int \frac{1}{\cos^2 x \sin ^2 x} dx\)
    
15. \( \int \sqrt{1-\cos px } dx\)
    
16. \( \int \frac{1}{1+\cos m x} dx\)
    
17. \( \int \frac{1}{1-\sin a x} dx\)
    
18. \( \int \sin 6 x \cos 8x dx\)
    
19. \( \int \sin^7 x \sin ^5 x dx\)
    
20. \( \int \cos px \cos q x dx\)
    
21. \( \int \frac{\cos x- \cos 2x}{1-\cos x} dx\)

    Solution 👉 Click Here

    Given integral is
    \( \int \frac{\cos x- \cos 2x}{1-\cos x} dx \)
    or\( \int \frac{\cos x- (2 \cos^2x -1)}{1-\cos x} dx\)
    or\( \int \frac{2 \cos^2x - \cos x-1)}{\cos x-1} dx \)
    or\(\int \frac{(2 \cos x-1) (\cos x-1)}{\cos x-1} dx \)
    or\(\int (2 \cos x-1) dx \)
    or\(2 \sin x-x+c \)

Basic Integral of Logarithm functions: Activity 5

Compute the following integrals.

1. \( \int (e^{px}+e^{-qx})dx\)
   
2. \( \int (e^{px}+e^{-px})^2 dx\)
   
3. \( \int \frac{e^{2x}+e^x+1}{e^x} dx\)
   
4. \( \int e^x (e^{2x}+1) dx\)
   
5. \( \int \frac{e^{ 6\log x}-e^{ 5\log x}}{e^{ 4\log x}-e^{ 3\log x}} dx\)

   Solution 👉 Click Here

   यस प्रश्नमा \(e^{\log x} =x \) मानौ ।
   Given integral is
   \( \int \frac{e^{ 6\log x}-e^{ 5\log x}}{e^{ 4\log x}-e^{ 3\log x}} dx \)
   or\( \int \frac{e^{ \log x ^6}-e^{ \log x^5}}{e^{ \log x^4}-e^{ \log x^3}} dx\)
   or\( \int \frac{x^6-x^5}{x^4-x^3} dx \)
   or\( \int x^2 dx \)
   or\( \frac{x^3}{3}+c \) \(\int x^n dx=\frac{x^{n+1}}{n+1}\)
   or\( \frac{1}{3}e^{\log x^3}+c \)
   or\( \frac{1}{3}e^{3\log x}+c \)