# 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.