# Integration by Substitution

Tutorials with examples and detailed solutions and exercises with answers on how to use the powerful technique of integration by substitution to find integrals.

## Review Integration by Substitution

The method of integration by substitution may be used to easily compute complex integrals. Let us examine an integral of the form_{a}

^{b}f(g(x)) g'(x) dx

With the above substitution, the given integral is given by

_{a}

^{b}f(g(x)) g'(x) dx =

_{g(a)}

^{g(b)}f(u) du

In what follows C is a constant of integration which is added in the final result.

## Examples
## Example 1Evaluate the integral
## Example 2Evaluate the integral^{3x - 2} dx
## Example 3Evaluate the integral^{2} + 5)^{4} dx
## Example 4Evaluate the integral
## Example 5Evaluate the integral^{-4} dx
## Example 6Evaluate the integral^{x2 + 2} dx
## Example 7Evaluate the integral^{4}(x) dx
## Example 8Evaluate the integral
## Example 9Evaluate the integral
## Example 10Evaluate the integral^{3}(x + 4)^{2} dx
## Example 11Evaluate the integral^{2} + 3x + 1) ) dx
## ExercisesUse the table of integrals and the method of integration by parts to find the integrals below. [Note that you may need to use the method of integration by parts more than once].1. cos(3x - 2) dx 2. e ^{4x - 7} dx
3. x(4x ^{2} + 5)^{4} dx
4. 1 / (x + 3) ^{3} dx
## Answers to Above Exercises1. (1/3) sin(3x - 2) 2. (1/4) e ^{4x - 7} + C
3. (1/40) (4x ^{2} + 5)^{5} + C
4. (-1/2) 1 / (x + 3) ^{2} + C
## More References and linksintegrals and their applications in calculus. |