# Integrate (2+x)^2-1x^2^2 from -1 to 2

## \int_{-1}^{2}\left(\left(2+x\right)^2-\left(x^2\right)^2\right)dx\cdot\frac{1}{2}

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$\frac{36}{5}$

## Step by step solution

Problem

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

Applying the power of a power property

$\frac{1}{2}\int_{-1}^{2}\left(\left(x+2\right)^2-x^{4}\right)dx$
2

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

$\frac{1}{2}\left(\int_{-1}^{2}-x^{4}dx+\int_{-1}^{2}\left(x+2\right)^2dx\right)$
3

Taking the constant out of the integral

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx-\int_{-1}^{2} x^{4}dx\right)$
4

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

$\frac{1}{2}\left(\left[-\frac{x^{5}}{5}\right]_{-1}^{2}+\int_{-1}^{2}\left(x+2\right)^2dx\right)$
5

Evaluate the definite integral

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx-1\cdot \left(\frac{{\left(-1\right)}^{5}}{5}\right)\left(-1\right)+\frac{2^{5}}{5}\left(-1\right)\right)$
6

Multiply $-1$ times $-1$

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx+\frac{{\left(-1\right)}^{5}}{5}\cdot 1+\frac{2^{5}}{5}\left(-1\right)\right)$
7

Calculate the power

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx+\frac{-1}{5}\cdot 1+\frac{32}{5}\left(-1\right)\right)$
8

Divide $32$ by $5$

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx-\frac{1}{5}\cdot 1+\frac{32}{5}\left(-1\right)\right)$
9

Multiply $-1$ times $\frac{32}{5}$

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx-\frac{1}{5}-\frac{32}{5}\right)$
10

Subtract the values $-\frac{1}{5}$ and $-\frac{32}{5}$

$\frac{1}{2}\left(\int_{-1}^{2}\left(x+2\right)^2dx-\frac{33}{5}\right)$
11

Apply the formula: $\int\left(a+x\right)^ndx$$=\frac{\left(a+x\right)^{\left(1+n\right)}}{1+n}$, where $a=2$ and $n=2$

$\frac{1}{2}\left(\left[\frac{\left(2+x\right)^{3}}{3}\right]_{-1}^{2}-\frac{33}{5}\right)$
12

Evaluate the definite integral

$\frac{1}{2}\cdot \left(-6.6+\frac{\left(2-1\right)^{3}}{3}\left(-1\right)+\frac{\left(2+2\right)^{3}}{3}\right)$
13

Subtract the values $2$ and $-1$

$\frac{1}{2}\cdot \left(-6.6+\frac{1^{3}}{3}\left(-1\right)+\frac{\left(2+2\right)^{3}}{3}\right)$
14

Add the values $2$ and $2$

$\frac{1}{2}\cdot \left(-6.6+\frac{1^{3}}{3}\left(-1\right)+\frac{4^{3}}{3}\right)$
15

Calculate the power

$\frac{1}{2}\cdot \left(-6.6+\frac{1}{3}\left(-1\right)+\frac{64}{3}\right)$
16

Divide $64$ by $3$

$\frac{1}{2}\cdot \left(-6.6+0.3333\left(-1\right)+21.3333\right)$
17

Subtract the values $\frac{64}{3}$ and $-\frac{33}{5}$

$\frac{1}{2}\cdot \left(0.3333\left(-1\right)+14.7333\right)$
18

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

$\frac{1}{2}\cdot \left(14.7333-0.3333\right)$
19

Subtract the values $14.7333$ and $-\frac{1}{3}$

$\frac{1}{2}\cdot \frac{72}{5}$
20

Multiply $\frac{72}{5}$ times $\frac{1}{2}$

$\frac{36}{5}$

$\frac{36}{5}$

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### Main topic:

Integration by substitution

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