Quantum Chemistry/Integration by change of variables

When computing integrals, an effective way to turn a complex integral of the form $\int f(g(x))g^{\prime }(x)dx$ 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, $u$ for $g(x)$. 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;

1. Choose your substitution, $u=g(x)$: Identify which part of the function to be integrated you'd like to substitute for the variable $u$. The substituted portion should have a derivative, $du$, which is similar to (ie. a multiple of) the rest of the integral.
2. Compute the derivative, $\frac{du}{dx}=g^{\prime }(x)$: Find the derivative of the chosen part of the function with respect to $x$, then rearrange the expression to express the derivative of your new variable, $du$ in terms of $dx$, $du=g^{\prime }(x)dx$.
3. Substitution of the new variable: Replace the original function in terms of $x$ and $dx$ with your new function in terms of $u$ and $du$. This step should work to simplify the integral by replacing a part of the original function with $u$ and replacing $dx$ with a multiple of $du$.
4. Compute your new integral: Solve the new, simplified integral in terms of the substituted variable, $u$.
5. Substitution of the original variable: Replace the new variable with the original function, $u=g(x)$.

Example

$\int \sin (x)\cos (x)dx$

1. Choose your substitution, $u=g(x)$. In this case, you should notice that the derivative of $g(x)=\sin (x)$ with respect to $x$ gives a multiple of the rest of the integral. The most effective substitution for this case is therefore,

$u=g(x)=\sin (x)$

2. Compute the derivative, $\frac{du}{dx}=g^{\prime }(x)$. In this case, where $u=\sin (x)$, you would use the known derivative via trigonometric identities to compute the derivative. Then the derivative should be rearranged to give $du$ in terms of $dx$;

$\frac{du}{dx}=g^{\prime }(x)=\cos (x)$

$du=\cos (x)dx$

3. Substitution of the new variable: In this case, the variable $u$ will replace $\sin (x)$, and $du$ will replace $\cos (x)dx$.

$\int \sin (x)\cos (x)dx=\int udu$

4. Compute your new integral: In this case, the integral is solved by using the power rule.

$\int udu=\frac{u^2}{2}+C$

5. Substitution of the original variable: Substitute $u=g(x)=\sin (x)$ back into the function.

$\frac{u^2}{2}+C=\frac{{\sin }^2(x)}{2}+C$