# Integral of x^2(9-1x^2)^0.5

## \intx^2\sqrt{9-x^2}dx

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$-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta+\frac{81}{2}arcsin\left(\frac{x}{3}\right)+\frac{9}{4}x\sqrt{9-x^2}-\frac{81}{4}arcsin\left(\frac{x}{3}\right)$

## Step by step solution

Problem

$\intx^2\sqrt{9-x^2}dx$
1

Solve the integral $\int\sqrt{9-x^2}x^2$ by trigonometric substitution using the substitution

$\begin{matrix}x=3\sin\left(\theta\right) \\ dx=3\cos\left(\theta\right)d\theta\end{matrix}$
2

Substituting in the original integral, we get

$\int27\cos\left(\theta\right)\sqrt{9-9\sin\left(\theta\right)^2}\sin\left(\theta\right)^2d\theta$
3

Applying a sine identity in order to reduce the exponent: $\displaystyle\sin(\theta)=\sqrt{\frac{1-\cos(2\theta)}{2}}$

$\int27\cos\left(\theta\right)\sqrt{9-9\sin\left(\theta\right)^2}\cdot\frac{1-\cos\left(2\theta\right)}{2}d\theta$
4

Taking the constant out of the integral

$27\int\cos\left(\theta\right)\sqrt{9-9\sin\left(\theta\right)^2}\cdot\frac{1-\cos\left(2\theta\right)}{2}d\theta$
5

Factor by the greatest common divisor $9$

$27\int\cos\left(\theta\right)\sqrt{9\left(1-\sin\left(\theta\right)^2\right)}\cdot\frac{1-\cos\left(2\theta\right)}{2}d\theta$
6

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

$27\int3\cos\left(\theta\right)\frac{1-\cos\left(2\theta\right)}{2}\sqrt{1-\sin\left(\theta\right)^2}d\theta$
7

Simplify the fraction

$27\int\frac{3}{2}\cos\left(\theta\right)\sqrt{1-\sin\left(\theta\right)^2}\left(1-\cos\left(2\theta\right)\right)d\theta$
8

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

$27\int\frac{3}{2}\cos\left(\theta\right)\sqrt{\cos\left(\theta\right)^2}\left(1-\cos\left(2\theta\right)\right)d\theta$
9

Applying the power of a power property

$27\int\frac{3}{2}\cos\left(\theta\right)\cos\left(\theta\right)\left(1-\cos\left(2\theta\right)\right)d\theta$
10

When multiplying exponents with same base you can add the exponents

$27\int\frac{3}{2}\left(1-\cos\left(2\theta\right)\right)\cos\left(\theta\right)^2d\theta$
11

Taking the constant out of the integral

$27\cdot \frac{3}{2}\int\left(1-\cos\left(2\theta\right)\right)\cos\left(\theta\right)^2d\theta$
12

Multiply $\frac{3}{2}$ times $27$

$\frac{81}{2}\int\left(1-\cos\left(2\theta\right)\right)\cos\left(\theta\right)^2d\theta$
13

Multiplying polynomials $\cos\left(\theta\right)^2$ and $1+-\cos\left(2\theta\right)$

$\frac{81}{2}\int\left(\cos\left(\theta\right)^2-\cos\left(\theta\right)^2\cos\left(2\theta\right)\right)d\theta$
14

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{81}{2}\left(\int-\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta+\int\cos\left(\theta\right)^2d\theta\right)$
15

Taking the constant out of the integral

$\frac{81}{2}\left(\int\cos\left(\theta\right)^2d\theta-\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta\right)$
16

Multiply $\left(\int\cos\left(\theta\right)^2d\theta+-\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta\right)$ by $\frac{81}{2}$

$\frac{81}{2}\int\cos\left(\theta\right)^2d\theta-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
17

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

$\frac{81}{2}\int\left(1-\sin\left(\theta\right)^2\right)d\theta-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
18

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{81}{2}\left(\int-\sin\left(\theta\right)^2d\theta+\int1d\theta\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
19

Applying a sine identity in order to reduce the exponent: $\displaystyle\sin(\theta)=\sqrt{\frac{1-\cos(2\theta)}{2}}$

$\frac{81}{2}\left(\int-\frac{1-\cos\left(2\theta\right)}{2}d\theta+\int1d\theta\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
20

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

$\frac{81}{2}\left(\int-\frac{1-\cos\left(2\theta\right)}{2}d\theta+\theta\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
21

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

$\frac{81}{2}\left(\int\frac{-\left(1-\cos\left(2\theta\right)\right)}{2}d\theta+arcsin\left(\frac{x}{3}\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
22

Taking the constant out of the integral

$\frac{81}{2}\left(\frac{1}{2}\int-\left(1-\cos\left(2\theta\right)\right)d\theta+arcsin\left(\frac{x}{3}\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
23

Taking the constant out of the integral

$\frac{81}{2}\left(\frac{1}{2}\left(-1\right)\int\left(1-\cos\left(2\theta\right)\right)d\theta+arcsin\left(\frac{x}{3}\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
24

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

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\int\left(1-\cos\left(2\theta\right)\right)d\theta\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
25

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(\int-\cos\left(2\theta\right)d\theta+\int1d\theta\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
26

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

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(\int-\cos\left(2\theta\right)d\theta+\theta\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
27

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

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(\int-\cos\left(2\theta\right)d\theta+arcsin\left(\frac{x}{3}\right)\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
28

Taking the constant out of the integral

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(arcsin\left(\frac{x}{3}\right)-\int\cos\left(2\theta\right)d\theta\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
29

Apply the formula: $\int\cos\left(x\cdot a\right)dx$$=\frac{1}{a}\sin\left(x\cdot a\right)$, where $a=2$ and $x=\theta$

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(arcsin\left(\frac{x}{3}\right)-1\cdot \frac{1}{2}\sin\left(2\theta\right)\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
30

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

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\sin\left(2\theta\right)\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
31

Using the sine double-angle identity

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\cdot 2\cos\left(\theta\right)\sin\left(\theta\right)\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
32

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

$\frac{81}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{1}{2}\left(arcsin\left(\frac{x}{3}\right)-\frac{x\sqrt{9-x^2}}{9}\right)\right)-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta$
33

Multiply $\left(arcsin\left(\frac{x}{3}\right)+-\frac{x\sqrt{9-x^2}}{9}\right)$ by $-\frac{81}{4}$

$-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta+\frac{81}{2}arcsin\left(\frac{x}{3}\right)+\frac{81}{4}\cdot\frac{x\sqrt{9-x^2}}{9}-\frac{81}{4}arcsin\left(\frac{x}{3}\right)$
34

Simplify the fraction

$-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta+\frac{81}{2}arcsin\left(\frac{x}{3}\right)+\frac{9}{4}x\sqrt{9-x^2}-\frac{81}{4}arcsin\left(\frac{x}{3}\right)$

$-\frac{81}{2}\int\cos\left(\theta\right)^2\cos\left(2\theta\right)d\theta+\frac{81}{2}arcsin\left(\frac{x}{3}\right)+\frac{9}{4}x\sqrt{9-x^2}-\frac{81}{4}arcsin\left(\frac{x}{3}\right)$

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### Main topic:

Integration by trigonometric substitution

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