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:

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