Integrate (1+x^2)^0.5 from -1 to 1

\int_{-1}^{1}\sqrt{1+x^2}dx

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Answer

$\sqrt{5}$

Step by step solution

Problem

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

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

$\begin{matrix}x=\tan\left(\theta\right) \\ dx=\sec\left(\theta\right)^2d\theta\end{matrix}$
2

Substituting in the original integral, we get

$\int_{-1}^{1}\sec\left(\theta\right)^2\sqrt{\tan\left(\theta\right)^2+1}d\theta$
3

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

$\int_{-1}^{1}\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
4

When multiplying exponents with same base you can add the exponents

$\int_{-1}^{1}\sec\left(\theta\right)^{3}d\theta$
5

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}\right)$
6

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

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+\frac{\frac{x\left(\sqrt{x^2+1}\right)^{2}}{\sqrt{x^2+1}}}{2}\right)$
7

Applying the power of a power property

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+\frac{\frac{x\left(x^2+1\right)}{\sqrt{x^2+1}}}{2}\right)$
8

Simplifying the fraction by $x^2+1$

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+\frac{\frac{x}{\frac{1}{\sqrt{x^2+1}}}}{2}\right)$
9

Simplifying the fraction

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+\frac{x}{2\left(\frac{1}{\sqrt{x^2+1}}\right)}\right)$
10

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=2$ and $x=\sqrt{x^2+1}$

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+\frac{x}{\frac{2}{\sqrt{x^2+1}}}\right)$
11

Simplifying the fraction

$\left(\frac{1}{2}\int_{-1}^{1}\sec\left(\theta\right)d\theta+x\frac{\sqrt{x^2+1}}{2}\right)$
12

The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$

$\left[\left(\frac{1}{2}\ln\left(\tan\left(\theta\right)+\sec\left(\theta\right)\right)+x\frac{\sqrt{x^2+1}}{2}\right)\right]_{-1}^{1}$
13

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

$\left[\left(\frac{1}{2}\ln\left(x+\sqrt{x^2+1}\right)+x\frac{\sqrt{x^2+1}}{2}\right)\right]_{-1}^{1}$
14

Evaluate the definite integral

$\left(\ln\left(\sqrt{{\left(-1\right)}^2+1}-1\right)\cdot 0.5+\frac{\sqrt{{\left(-1\right)}^2+1}}{2}\left(-1\right)\right)\left(-1\right)+\ln\left(1+\sqrt{1^2+1}\right)\cdot 0.5+\frac{\sqrt{1^2+1}}{2}\cdot 1$
15

Calculate the power

$\left(\ln\left(\sqrt{1+1}-1\right)\cdot 0.5+\frac{\sqrt{1+1}}{2}\left(-1\right)\right)\left(-1\right)+\ln\left(1+\sqrt{1+1}\right)\cdot 0.5+\frac{\sqrt{1+1}}{2}\cdot 1$
16

Add the values $1$ and $1$

$\left(\ln\left(\sqrt{2}-1\right)\cdot 0.5+\frac{\sqrt{2}}{2}\left(-1\right)\right)\left(-1\right)+\ln\left(1+\sqrt{2}\right)\cdot 0.5+\frac{\sqrt{2}}{2}\cdot 1$
17

Calculate the square root of $2$

$\left(\ln\left(1.4142-1\right)\cdot 0.5+\frac{1.4142}{2}\left(-1\right)\right)\left(-1\right)+\ln\left(1+1.4142\right)\cdot 0.5+\frac{1.4142}{2}\cdot 1$
18

Subtract the values $\sqrt{2}$ and $-1$

$\left(\ln\left(0.4142\right)\cdot 0.5+\frac{1.4142}{2}\left(-1\right)\right)\left(-1\right)+\ln\left(1+1.4142\right)\cdot 0.5+\frac{1.4142}{2}\cdot 1$
19

Add the values $\sqrt{2}$ and $1$

$\left(\ln\left(0.4142\right)\cdot 0.5+\frac{1.4142}{2}\left(-1\right)\right)\left(-1\right)+\ln\left(2.4142\right)\cdot 0.5+\frac{1.4142}{2}\cdot 1$
20

Divide $\sqrt{2}$ by $2$

$\left(\ln\left(0.4142\right)\cdot 0.5+0.7071\left(-1\right)\right)\left(-1\right)+\ln\left(2.4142\right)\cdot 0.5+0.7071\cdot 1$
21

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

$\left(\ln\left(0.4142\right)\cdot 0.5-0.7071\right)\left(-1\right)+\ln\left(2.4142\right)\cdot 0.5+0.7071$
22

Calculating the natural logarithm of $\sqrt[3]{33}$

$\left(-0.7071-0.8814\cdot 0.5\right)\left(-1\right)+0.8814\cdot 0.5+0.7071$
23

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

$\left(-0.7071-0.4407\right)\left(-1\right)+0.4407+0.7071$
24

Subtract the values $-\frac{\sqrt{2}}{2}$ and $-\frac{26}{59}$

$-1.1478\left(-1\right)+0.4407+0.7071$
25

Add the values $\frac{\sqrt{2}}{2}$ and $\frac{26}{59}$

$1.1478-1.1478\left(-1\right)$
26

Multiply $-1$ times $-\frac{\sqrt{5}}{2}$

$1.1478+1.1478$
27

Add the values $\frac{\sqrt{5}}{2}$ and $\frac{\sqrt{5}}{2}$

$\sqrt{5}$

Answer

$\sqrt{5}$