1
Solved example of Integration by trigonometric substitution
$\int\sqrt{x^2-4}dx$
2
Solve the integral $\int\sqrt{x^2-4}dx$ by trigonometric substitution using the substitution
$\begin{matrix}x=2\sec\left(\theta\right) \\ dx=\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta\end{matrix}$
3
Substituting in the original integral, we get
$\int\sqrt{4\sec\left(\theta\right)^2-4}\cdot\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta$
4
Factor by the greatest common divisor $4$
$\int\sqrt{4\left(\sec\left(\theta\right)^2-1\right)}\cdot\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta$
5
The power of a product is equal to the product of it's factors raised to the same power
$\int2\sqrt{\sec\left(\theta\right)^2-1}\cdot\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta$
6
Applying the trigonometric identity: $\tan\left(\theta\right)^2=\sec\left(\theta\right)^2-1$
$\int2\tan\left(\theta\right)\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta$
7
Take the constant out of the integral
$2\int\tan\left(\theta\right)\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta$