Integration by-parts

$\int u dv = uv - \int v du$

The key is to find these expressions:

1. $du$: So you need to find a $u$ that can be easily differentiated
2. $v$: So you need to find a $dv$ that can be easily integrated

Example:

$E(X) = \int_{0}^{w} x f(x) dx$

Assign $u$ as $x$ and $dv$ as $f(x)$, then we have:

1. $\frac{du}{dx} = 1 \implies du = dx$
2. $\int dv = v \implies \int f(x) dx = F(x)$

Then we have: $E(X) = \int_{0}^{w} x f(x) dx = xF(x) - \int_{0}^{w} F(x) dx$

Even quicker:

Notice that you can rewrite $\int_{0}^{w} x f(x) dx = \int_{0}^{w} x dF(x)$

Why? Because $\frac{dF(x)}{dx} = f(x)$ or $dF(x) = f(x) dx$

Then you can apply integration by-parts directly with substituting $u = x$ and $v = F(x)$ in the original expression:

$\int_{0}^{w} x dF(x) = x F(x) - \int_{0}^{w} F(x) dx$