Integral of 1/((36-9x^2)^0.5)

\int\frac{1}{\sqrt{36-9x^2}}dx

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Answer

$\frac{1}{3}arcsin\left(\frac{3}{2}x\right)+C_0$

Step by step solution

Problem

$\int\frac{1}{\sqrt{36-9x^2}}dx$
1

First, factor the terms inside the radical for an easier handling

$\int\frac{1}{\sqrt{9\left(4-x^2\right)}}dx$
2

Taking the constant out of the radical

$\int\frac{1}{3\sqrt{4-x^2}}dx$
3

Solve the integral $\int\frac{1}{3\sqrt{4-x^2}}$ by trigonometric substitution using the substitution

$\begin{matrix}x=2\sin\left(\theta\right) \\ dx=2\cos\left(\theta\right)d\theta\end{matrix}$
4

Substituting in the original integral, we get

$\int\frac{2\cos\left(\theta\right)}{3\sqrt{4-4\sin\left(\theta\right)^2}}d\theta$
5

Factor by the greatest common divisor $4$

$\int\frac{2\cos\left(\theta\right)}{3\sqrt{4\left(1-\sin\left(\theta\right)^2\right)}}d\theta$
6

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

$\int\frac{2\cos\left(\theta\right)}{6\sqrt{1-\sin\left(\theta\right)^2}}d\theta$
7

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

$\int\frac{2\cos\left(\theta\right)}{6\sqrt{\cos\left(\theta\right)^2}}d\theta$
8

Applying the power of a power property

$\int\frac{2\cos\left(\theta\right)}{6\cos\left(\theta\right)}d\theta$
9

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

$\int\frac{2}{6}d\theta$
10

Divide $2$ by $6$

$\int0.3333d\theta$
11

The integral of a constant is equal to the constant times the integral's variable

$\frac{1}{3}\theta$
12

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

$\frac{1}{3}arcsin\left(\frac{3x}{6}\right)$
13

Apply the formula: $\frac{b\cdot a}{c}$$=b\frac{a}{c}$, where $a=3$, $b=x$ and $c=6$

$\frac{1}{3}arcsin\left(3\cdot \frac{1}{2}x\right)$
14

Multiply $\frac{1}{2}$ times $3$

$\frac{1}{3}arcsin\left(\frac{3}{2}x\right)$
15

Add the constant of integration

$\frac{1}{3}arcsin\left(\frac{3}{2}x\right)+C_0$

Answer

$\frac{1}{3}arcsin\left(\frac{3}{2}x\right)+C_0$

Problem Analysis

Main topic:

Integration by trigonometric substitution

Time to solve it:

0.27 seconds

Views:

113