Quantum Chemistry/Integration by change of variables

When computing integrals, an effective way to turn a complex integral of the form into a more simple function is by using the technique of "change of variables" (or substitution). This technique works by simplifying an integral by introducing a new variable, for . This method is most effective when the integral contains a function with known derivatives, or easy to work out derivatives as this process utilizes the chain rule. The general steps for the change of variables technique are as follows; Choose your substitution, : Identify which part of the function to be integrated you'd like to substitute for the variable . The substituted portion should have a derivative, , which is similar to (ie. a multiple of) the rest of the integral. Compute the derivative, : Find the derivative of the chosen part of the function with respect to , then rearrange the expression to express the derivative of your new variable, in terms of , . Substitution of the new variable: Replace the original function in terms of and with your new function in terms of and . This step should work to simplify the integral by replacing a part of the original function with and replacing with a multiple of . Compute your new integral: Solve the new, simplified integral in terms of the substituted variable, . Substitution of the original variable: Replace the new variable with the original function, . Example 1. Choose your substitution, . In this case, you should notice that the derivative of with respect to gives a multiple of the rest of the integral. The most effective substitution for this case is therefore, 2. Compute the derivative, . In this case, where , you would use the known derivative via trigonometric identities to compute the derivative. Then the derivative should be rearranged to give in terms of ; 3. Substitution of the new variable: In this case, the variable will replace , and will replace . 4. Compute your new integral: In this case, the integral is solved by using the power rule. 5. Substitution of the original variable: Substitute back into the function.