Integration by Substitution

${\displaystyle \int f(u){\frac {du}{dx}}dx=\int f(u)du}$

Integration by Parts

${\displaystyle \int u{\frac {dv}{dx}}dx=vu-\int v{\frac {du}{dx}}dx}$

Integration by parts is used when you have two functions multiplied together, such as ln x and a simple polynomial, where 1 function is not the derivative of the other. As an example:

${\displaystyle \int x\ln xdx}$

In this expression use the substitutions: ${\displaystyle u=\ln x}$ and ${\displaystyle {\frac {dv}{dx}}=x}$. In almost all other expressions, the polynomial is taken as u. After substituting, the expression in the example becomes:

${\displaystyle \int x\ln xdx=\ln x\int xdx-\int x{\frac {d}{dx}}\ln xdx}$

After integrating and differentiating the respective parts of the expression, this becomes:

${\displaystyle \int x\ln xdx={\frac {1}{2}}x^{2}\ln x-x.}$