Integration as inverse operator

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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

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Formula for Integral

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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	\( \frac{d}{dx} (ax+b)^n=na (ax+b)^{n-1}\)	\( \int (ax+b)^n dx= \frac{(ax+b)^{n+1}}{a(n+1)}+c ;n \ne -1\)

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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	\( \frac{d}{dx} \log |ax+b| = \frac{a}{ax+b}\)	\( \int \frac{dx}{ax+b} dx= \frac{1}{a} \log |ax+b|+c\)

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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	\( \frac{d}{dx} \left ( \csc x\right) = -\csc x \cot x\)	\( \int \left ( \csc x \cot x \right) dx= -\csc x+c \)
		\( \int \left ( \csc x \right) dx= -\log |\csc x+\cot x|+c \)

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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	\( \frac{d}{dx} a^{x} = a^{x} \log a\)	\( \int a^{x} dx=\frac{a^{x}}{\log a} +c\)

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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	\( \frac{d}{dx} \left ( \csc ^{-1} x\right) = \frac{-1}{|x| \sqrt{x^2-1}} \)	\( \int \left ( \frac{-1}{|x| \sqrt{x^2-1}} \right) dx=\csc ^{-1} x+c \)

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Basic Integral of algebraic functions: Activity 1

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Compute the following integrals.

1. \( \int 60 dt\)

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

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   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. \( \int 5x^3 dx\)

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   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. \( \int 7x^{5/2}dx\)

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   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. \( \int 4x^{-5} dx\)

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   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. \( \int 2x^{-\frac{7}{2}} dx \)

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   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. \( \int (w^2-3)dw \)

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   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. \( \int (x^2+2) dx\)

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   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. \( \int (2x+1)(3x+2) dx\)

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    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. \( \int (4x^{1/3}+5x^{2/3}+\frac{1}{x^2}) dx \)

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    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. \( \int (x^{3/4}+x^{1/2}+4x^{1/3} )dx\)

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    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. \( \int (x^2-\frac{1}{x^2}) dx\)

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    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. \( \int \sqrt{x} (x^2-5) dx\)

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

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

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    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. \( \int (x-3)^2 dx\)

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    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. \( \int (4x+5)^4dx \)

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    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. \( \int (3x+5)^4 dx\)

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    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. \( \int (a-bx)^5 dx\)

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    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. \( \int (c+dx)^{-3/2}dx\)

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    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. \( \int \frac{1}{\sqrt{2x+7} }dx\)

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    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\( 4x - \frac{1}{5(5x+1)}+c \)

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Basic Integral of algebraic functions: Activity 2

Solution 👉 Click Here

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\( \frac{4}{27} (3x+2)^{3/2} -\frac{2}{9} (3x+2)^{1/2}+c \)

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Basic Integral of algebraic functions: Activity 3

Solution 👉 Click Here

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

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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\(2 \sin x-x+c \)

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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\( \frac{1}{3}e^{3\log x}+c \)
   

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