Integral of 1/(x^2(4-1x^2)^0.5)

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

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Answer

$\frac{-\frac{1}{4}\sqrt{4-x^2}}{x}+C_0$

Step by step solution

Problem

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

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

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

Substituting in the original integral, we get

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

Factor by the greatest common divisor $4$

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

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

$\int\frac{2\cos\left(\theta\right)}{8\sin\left(\theta\right)^2\sqrt{1-\sin\left(\theta\right)^2}}d\theta$
5

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

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

Applying the power of a power property

$\int\frac{2\cos\left(\theta\right)}{8\sin\left(\theta\right)^2\cos\left(\theta\right)}d\theta$
7

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

$\int\frac{2}{8\sin\left(\theta\right)^2}d\theta$
8

Taking the constant out of the integral

$\frac{1}{8}\int\frac{2}{\sin\left(\theta\right)^2}d\theta$
9

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$\frac{1}{8}\int2\csc\left(\theta\right)^2d\theta$
10

Taking the constant out of the integral

$\frac{1}{8}\cdot 2\int\csc\left(\theta\right)^2d\theta$
11

Multiply $2$ times $\frac{1}{8}$

$\frac{1}{4}\int\csc\left(\theta\right)^2d\theta$
12

The integral of $\csc(x)^2$ is $-\cot(x)$

$\frac{1}{4}\left(-1\right)\cot\left(\theta\right)$
13

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

$\frac{-\frac{1}{4}\sqrt{4-x^2}}{x}$
14

Add the constant of integration

$\frac{-\frac{1}{4}\sqrt{4-x^2}}{x}+C_0$

Answer

$\frac{-\frac{1}{4}\sqrt{4-x^2}}{x}+C_0$

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Problem Analysis

Main topic:

Integration by trigonometric substitution

Time to solve it:

0.29 seconds

Views:

101