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.

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’.

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.

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.

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.