Integrate ((-2x)/(2(9-1x^2)^0.5))^2^0.5 from -1 to 2

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

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Answer

$0$

Step by step solution

Problem

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

Applying the power of a power property

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

Taking the constant out of the integral

$-2\int_{-1}^{2}\frac{x}{2\sqrt{9-x^2}}dx$
3

Taking the constant out of the integral

$-2\cdot \frac{1}{2}\int_{-1}^{2}\frac{x}{\sqrt{9-x^2}}dx$
4

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

$-\int_{-1}^{2}\frac{x}{\sqrt{9-x^2}}dx$
5

Solve the integral $\int\frac{x}{\sqrt{9-x^2}}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=9-x^2 \\ du=-2xdx\end{matrix}$
6

Isolate $dx$ in the previous equation

$\frac{du}{-2x}=dx$
7

Substituting $u$ and $dx$ in the integral

$-\int_{-1}^{2}\frac{1}{-2\sqrt{u}}du$
8

Since the integral has a discontinuity inside the interval, we have to split it in two integrals

$-\int_{-1}^{0}\frac{1}{-2\sqrt{u}}du-\int_{0}^{2}\frac{1}{-2\sqrt{u}}du$
9

Replace the integral's limit by a finite value

$\lim_{c\to0}\:-\int_{c}^{2}\frac{1}{-2\sqrt{u}}du-\int_{-1}^{0}\frac{1}{-2\sqrt{u}}du$
10

Replace the integral's limit by a finite value

$\lim_{c\to0}\:-\int_{c}^{2}\frac{1}{-2\sqrt{u}}du+\lim_{c\to0}\:-\int_{-1}^{c}\frac{1}{-2\sqrt{u}}du$
11

Taking the constant out of the integral

$\lim_{c\to0}\:-1\left(-\frac{1}{2}\right)\int_{c}^{2}\frac{1}{\sqrt{u}}du+\lim_{c\to0}\:-\int_{-1}^{c}\frac{1}{-2\sqrt{u}}du$
12

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

$\lim_{c\to0}\:\frac{1}{2}\int_{c}^{2}\frac{1}{\sqrt{u}}du+\lim_{c\to0}\:-\int_{-1}^{c}\frac{1}{-2\sqrt{u}}du$
13

Taking the constant out of the integral

$\lim_{c\to0}\:\frac{1}{2}\int_{c}^{2}\frac{1}{\sqrt{u}}du+\lim_{c\to0}\:-1\left(-\frac{1}{2}\right)\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
14

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

$\lim_{c\to0}\:\frac{1}{2}\int_{c}^{2}\frac{1}{\sqrt{u}}du+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
15

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\lim_{c\to0}\:\frac{1}{2}\int_{c}^{2} u^{-\frac{1}{2}}du+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
16

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\lim_{c\to0}\left(\left[\frac{1}{2}\cdot 2\sqrt{u}\right]_{c}^{2}\right)+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
17

Substitute $u$ back for it's value, $9-x^2$

$\lim_{c\to0}\left(\left[1\sqrt{9-x^2}\right]_{c}^{2}\right)+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c} u^{-\frac{1}{2}}du$
18

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\lim_{c\to0}\left(\left[1\sqrt{9-x^2}\right]_{c}^{2}\right)+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
19

Any expression multiplied by $1$ is equal to itself

$\lim_{c\to0}\left(\left[\sqrt{9-x^2}\right]_{c}^{2}\right)+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
20

Evaluate the definite integral

$\lim_{c\to0}\left(\sqrt{9-x^2}-\sqrt{9-x^2}\right)+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
21

Subtracting $\sqrt{9-x^2}$ and $\sqrt{9-x^2}$

$\lim_{c\to0}\left(0\right)+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
22

The limit of a constant is just the constant

$0+\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
23

$x+0=x$, where $x$ is any expression

$\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c}\frac{1}{\sqrt{u}}du$
24

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\lim_{c\to0}\:\frac{1}{2}\int_{-1}^{c} u^{-\frac{1}{2}}du$
25

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\lim_{c\to0}\left(\left[\frac{1}{2}\cdot 2\sqrt{u}\right]_{-1}^{c}\right)$
26

Substitute $u$ back for it's value, $9-x^2$

$\lim_{c\to0}\left(\left[1\sqrt{9-x^2}\right]_{-1}^{c}\right)$
27

Any expression multiplied by $1$ is equal to itself

$\lim_{c\to0}\left(\left[\sqrt{9-x^2}\right]_{-1}^{c}\right)$
28

Evaluate the definite integral

$\lim_{c\to0}\left(\sqrt{9-x^2}-\sqrt{9-x^2}\right)$
29

Subtracting $\sqrt{9-x^2}$ and $\sqrt{9-x^2}$

$\lim_{c\to0}\left(0\right)$
30

The limit of a constant is just the constant

$0$

Answer

$0$

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