A-level Mathematics/MEI/C3/Integration

Integration

Integration by Substitution

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

Integration by Parts

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

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:

$\int x\ln xdx$

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

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

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

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