Integration by substitution

In calculus, Integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians.

This method is actually a reverse process of the chain rule. In fact, it is based on the chain rule, and you choose u based on the same criteria as you do in the chain rule.

The working of substitution method for indefinite integral

• Write down the given integral

• Come up with a substitution u = u(x).
• Ideally you may want to find the inverse function of u(x), meaning that you will find x = x(u).
• Differentiate to find dx = x'(u) du.
• Back to the given integral and make the appropriate substitutions

• Check after algebraic simplifications that the new integral is easier than the initial one. Otherwise, go back to step 2 and come up with another substitution.
• Do not forget that the answer to is a function of x. Therefore once you have finished doing all your calculations, you should substitute back to the initial variable x.

The steps for doing integration by substitution for definite integrals are the same as the steps for integration by substitution for indefinite integrals except we must change the bounds of integration and we do not need to sub back in for u.

There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable and the integrand. When dealing with definite integrals, the limits of integration can also change.

It is possible to transform a difficult integral to an easier integral by using a substitution.

For example, suppose we are integrating a difficult integral which is with respect to x. We might be able to let x = sin t, say, to make the integral easier. As long as we change "dx" to "cos t dt" (because if x = sin t then dx/dt = cost) we can now integrate with respect to t and we will get the same answer as if we had done the original integral.

The second one is especially important. If you want to integrate a fraction, where the top is the differential of the bottom, the answer is simply ln of the bottom plus a constant.