A-level Mathematics/MEI/C3/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:

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