# Integration by Substitution

# Integration by Substitution

## Definition

**Integration by substitution**is a method used in integral calculus to simplify certain integrals, making them easier to solve.- It’s analogous to the
**chain rule**for differentiation but applied in reverse. - The technique involves substituting a part of the given function with a new variable, which simplifies the integral.

## Technique

- To implement integration by substitution, we first identify a function and its derivative within the integral.
- The chosen function is replaced with a new variable, usually denoted by ‘u’, hence the name ‘u-substitution’.
- The original integral is then expressed in terms of ‘u’, which simplifies it into a standard integral form that can be integrated more straightforwardly.
- After performing the integration in terms of ‘u’, the last step involves replacing ‘u’ with the original function to obtain the integral in terms of ‘x’.

## Example of Applying Integration by Substitution

- Consider the integral ∫2x(x^2 + 1)^3 dx; we can let u= x^2+1 making du=2x dx, turning the integral into ∫u^3 du, which is easier to integrate.
- After performing the integration, we replace ‘u’ with the original function to find the original integral in terms of ‘x’, ∫2x(x^2 + 1)^3 dx = 1/4*(x^2 +1)^4+C.

## Common Mistakes in Integration by Substitution

- Not choosing the correct function to substitute, which ends up complicating the integral rather than simplifying it.
- Forgetting to replace ‘u’ with the original function after performing the integration, thus not correctly answering the question.

## Applications of Integration by Substitution

- Integration by substitution is fundamental in evaluating integrals that may not have seemed tractable in the first instance.
- It’s critical for tackling
**integral calculus problems**, including those that appear in engineering, physics, and numerous other scientific fields. - This technique also forms the foundation of more sophisticated techniques such as
**integration by parts**and**partial fractions**.