Integrating Using the Chain Rule Backwards

The chain rule in calculus can be applied backwards as a technique called integration by substitution for solving complex integrals.
The method involves substitifying a part of the integrand with a new variable for simplifying the integration.
To apply this method, initially a function ‘u’ inside the integrand is identified that, when replaced, will simplify the integral.
Then, derivative of this ‘u’ function is found with respect to ‘x’ (du/dx).
The integral is rewritten in terms of new variable ‘u’ and the integration is performed.
Finally, the original expression for ‘u’ is substituted back to bring back the original variable ‘x’.
The method of integration by substitution copies the chain rule in reverse and can solve various integrals that look intimidating initially.
Mastering this technique is crucial for progress in calculus, and it’s highly valuable in advanced mathematical coursework and applications across various disciplines.