Integral of 1/((x^2-36)^0.5)

\int\frac{1}{\sqrt{x^2-36}}dx

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Answer

$\ln\left(\frac{\sqrt{x^2-36}}{6}+\frac{x}{6}\right)+C_0$

Step by step solution

Problem

$\int\frac{1}{\sqrt{x^2-36}}dx$
1

Solve the integral $\int\frac{1}{\sqrt{x^2-36}}$ by trigonometric substitution using the substitution

$\begin{matrix}x=6\sec\left(\theta\right) \\ dx=6\tan\left(\theta\right)\sec\left(\theta\right)d\theta\end{matrix}$
2

Substituting in the original integral, we get

$\int\frac{6\tan\left(\theta\right)\sec\left(\theta\right)}{\sqrt{36\sec\left(\theta\right)^2-36}}d\theta$
3

Factor by the greatest common divisor $36$

$\int\frac{6\tan\left(\theta\right)\sec\left(\theta\right)}{\sqrt{36\left(\sec\left(\theta\right)^2-1\right)}}d\theta$
4

The power of a product is equal to the product of it's factors raised to the same power

$\int\frac{6\tan\left(\theta\right)\sec\left(\theta\right)}{6\sqrt{\sec\left(\theta\right)^2-1}}d\theta$
5

Simplifying the fraction by $6$

$\int\frac{\tan\left(\theta\right)\sec\left(\theta\right)}{\sqrt{\sec\left(\theta\right)^2-1}}d\theta$
6

Applying the trigonometric identity: $\tan\left(\theta\right)^2=\sec\left(\theta\right)^2-1$

$\int\frac{\tan\left(\theta\right)\sec\left(\theta\right)}{\tan\left(\theta\right)}d\theta$
7

Simplifying the fraction by $\tan\left(\theta\right)$

$\int\sec\left(\theta\right)d\theta$
8

The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$

$\ln\left(\tan\left(\theta\right)+\sec\left(\theta\right)\right)$
9

Expressing the result of the integral in terms of the original variable

$\ln\left(\frac{\sqrt{x^2-36}}{6}+\frac{x}{6}\right)$
10

Add fraction's numerators with common denominators: $\frac{x}{6}$ and $\frac{\sqrt{x^2-36}}{6}$

$\ln\left(\frac{\sqrt{x^2-36}+x}{6}\right)$
11

Split the fraction $\frac{x+\sqrt{x^2-36}}{6}$ in two terms with same denominator

$\ln\left(\frac{\sqrt{x^2-36}}{6}+\frac{x}{6}\right)$
12

Add the constant of integration

$\ln\left(\frac{\sqrt{x^2-36}}{6}+\frac{x}{6}\right)+C_0$

Answer

$\ln\left(\frac{\sqrt{x^2-36}}{6}+\frac{x}{6}\right)+C_0$

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Problem Analysis

Main topic:

Integration by trigonometric substitution

Time to solve it:

0.28 seconds

Views:

122