Integration by substitution is the inverse of the chain rule of differentiation.
The chain rule states that the derivative of a composite function, f(g(x)), is the product of the derivative of the outer function f(x) evaluated at g(x) and the derivative of the inner function g(x).
Integration by substitution essentially reverses this process. It allows us to simplify integrals by substituting a part of the integrand with a new variable, making the integral easier to solve.
Here's a simple example:
Let's say we want to find the integral of ∫2x(x^2 + 1) dx.
- We can use the substitution u = x^2 + 1.
- Then du = 2x dx.
- Substituting these into the integral, we get ∫ u du.
- This integral is much easier to solve, giving us (u^2)/2 + C.
- Finally, we substitute back u = x^2 + 1 to get the final answer: (x^2 + 1)^2 / 2 + C.
In essence, integration by substitution "undoes" the chain rule, allowing us to find the original function that was differentiated using the chain rule.
