Integration as inverse operator





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

Solution ๐Ÿ‘‰ Click Here




Basic Integral of algebraic functions: Activity 1

Solution ๐Ÿ‘‰ Click Here




Basic Integral of algebraic functions: Activity 2

Solution ๐Ÿ‘‰ Click Here




Basic Integral of algebraic functions: Activity 3

Solution ๐Ÿ‘‰ Click Here




Basic Integral of trigonometric functions: Activity 4

Compute the following integrals.
  1. \( \int \sin 5x dx\)

    Solution ๐Ÿ‘‰ Click Here

  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 ๐Ÿ‘‰ Click Here




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 ๐Ÿ‘‰ Click Here




No comments:

Post a Comment