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Integration by trigonometric substitution Calculator

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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

$2\int\tan\left(\theta\right)\frac{d}{d\theta}\left(2\sec\left(\theta\right)\right)d\theta$