The following note shows some integration techniques.
Antiderivatives
Let \(f:[a,b] \to \mathbb{R}\) be differentiable. Then
\[\int_{a}^{b} f'(x) \, d = f(b)-f(a) = f(x) \biggr\rvert_{a}^{b}.\]Examples
e
Integration by parts
Recall the integration by parts formula:
\[\int_{a}^{b} f'(x) g(x) \, dx = f(x)g(x) \biggr\rvert_{a }^{b } - \int_{a}^{b} g'(x) f(x) \, dx\]Examples
Change of variables
Recall that
\[\int_{a}^{b} f(\phi(t)) \phi'(t) \, dt = \int_{a}^{b} f(u) \, du,\]with \(u = \phi(t)\) and \(du = \phi'(t) dt\).
Examples
Trigonometric substitution
The following substitutions are usually helpful when these expressions appear.
\[\begin{aligned} \sqrt{ x^{2} -a^{2}} &\longrightarrow x = a \sec (\theta) \\ \sqrt{ x^{2}+a^{2} } &\longrightarrow x = a \tan (\theta) \\ \sqrt{ a^{2}-x^{2} } &\longrightarrow x = a \sin(\theta). \end{aligned}\]Examples
Trigonometric identities
The following formulas are usually helpful:
- \(\cos mx \sin nx = \frac{1}{2}(\sin((m+n)x)) - \sin((m-n)x))\).
- \(\cos mx \cos nx = \frac{1}{2} (\cos((m+n)x)+\cos((m-n)x))\).
- \(\sin mx \sin nx = \frac{1}{2}(\cos((m+n)x))-\cos((m-n)x)\).
Full list of identities
