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1

Solved example of integration by trigonometric substitution

$\int\sqrt{x^2+4}dx$
2

We can solve the integral $\int\sqrt{x^2+4}dx$ by applying integration method of trigonometric substitution using the substitution

$x=2\tan\left(\theta \right)$

Differentiate both sides of the equation $x=2\tan\left(\theta \right)$

$dx=\frac{d}{d\theta}\left(2\tan\left(\theta \right)\right)$

Find the derivative

$\frac{d}{d\theta}\left(2\tan\left(\theta \right)\right)$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

$2\frac{d}{d\theta}\left(\tan\left(\theta \right)\right)$

The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

$2\frac{d}{d\theta}\left(\theta \right)\sec\left(\theta \right)^2$

The derivative of the linear function is equal to $1$

$2\sec\left(\theta \right)^2$
3

Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above

$dx=2\sec\left(\theta \right)^2d\theta$

4

Substituting in the original integral, we get

$\int2\sqrt{4\tan\left(\theta \right)^2+4}\sec\left(\theta \right)^2d\theta$
5

Factor the polynomial $4\tan\left(\theta \right)^2+4$ by it's greatest common factor (GCF): $4$

$\int2\sqrt{4\left(\tan\left(\theta \right)^2+1\right)}\sec\left(\theta \right)^2d\theta$
6

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

$\int4\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
7

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

$\int4\sqrt{\sec\left(\theta \right)^2}\sec\left(\theta \right)^2d\theta$
8

The integral of a function times a constant ($4$) is equal to the constant times the integral of the function

$4\int\sqrt{\sec\left(\theta \right)^2}\sec\left(\theta \right)^2d\theta$
9

Simplify $\sqrt{\sec\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

$4\int\sec\left(\theta \right)\sec\left(\theta \right)^2d\theta$
10

When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)\sec\left(\theta \right)^2$

$4\int\sec\left(\theta \right)^{3}d\theta$
11

Rewrite $\sec\left(\theta \right)^{3}$ as the product of two secants

$4\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$
12

We can solve the integral $\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$

$\frac{d}{d\theta}\left(\theta \right)\sec\left(\theta \right)\tan\left(\theta \right)$

The derivative of the linear function is equal to $1$

$\sec\left(\theta \right)\tan\left(\theta \right)$
13

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\sec\left(\theta \right)}\\ \displaystyle{du=\sec\left(\theta \right)\tan\left(\theta \right)d\theta}\end{matrix}$
14

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sec\left(\theta \right)^2d\theta}\\ \displaystyle{\int dv=\int \sec\left(\theta \right)^2d\theta}\end{matrix}$
15

Solve the integral

$v=\int\sec\left(\theta \right)^2d\theta$
16

The integral of $\sec(x)^2$ is $\tan(x)$

$\tan\left(\theta \right)$

When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents

$4\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$
17

Now replace the values of $u$, $du$ and $v$ in the last formula

$4\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$
18

Multiply the single term $4$ by each term of the polynomial $\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$

$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$

Applying the trigonometric identity: $\tan^2(\theta)=\sec(\theta)^2-1$

$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta$
19

We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately

$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta$

$\int\left(\sec\left(\theta \right)^2\sec\left(\theta \right)-\sec\left(\theta \right)\right)$

When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)^2\sec\left(\theta \right)$

$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$
20

Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$

$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$
21

Expand the integral $\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$

Express the variable $\theta$ in terms of the original variable $x$

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

Multiplying the fraction by $4\left(\frac{\sqrt{x^2+4}}{2}\right)$

$\sqrt{x^2+4}x-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$
22

Express the variable $\theta$ in terms of the original variable $x$

$\sqrt{x^2+4}x-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$

Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$

Solve the product $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$

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

Simplify the fraction $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$

$-2\sin\left(\theta \right)\sec\left(\theta \right)^{2}-2\int\sec\left(\theta \right)d\theta$

Express the variable $\theta$ in terms of the original variable $x$

$-\frac{1}{2}\sqrt{x^2+4}x-2\int\sec\left(\theta \right)d\theta$

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

$-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$

Express the variable $\theta$ in terms of the original variable $x$

$-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
23

The integral $-4\int\sec\left(\theta \right)^{3}d\theta$ results in: $-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$

$-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
24

Gather the results of all integrals

$\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)-\frac{1}{2}\sqrt{x^2+4}x-4\int-\sec\left(\theta \right)d\theta$
25

Combining like terms $\sqrt{x^2+4}x$ and $-\frac{1}{2}\sqrt{x^2+4}x$

$\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)-4\int-\sec\left(\theta \right)d\theta$

The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function

$4\int\sec\left(\theta \right)d\theta$

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

$4\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$

Express the variable $\theta$ in terms of the original variable $x$

$4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
26

The integral $-4\int-\sec\left(\theta \right)d\theta$ results in: $4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$

$4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
27

Gather the results of all integrals

$\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)+4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
28

Combining like terms $-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$ and $4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$

$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
29

The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

$L.C.M.=2$
30

Combine and simplify all terms in the same fraction with common denominator $2$

$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\frac{\sqrt{x^2+4}+x}{2}\right)$
31

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\frac{\sqrt{x^2+4}+x}{2}\right)+C_0$
32

Simplify the expression by applying logarithm properties

$2\ln\left(\sqrt{x^2+4}+x\right)+\frac{1}{2}\sqrt{x^2+4}x+C_1$

Final Answer

$2\ln\left(\sqrt{x^2+4}+x\right)+\frac{1}{2}\sqrt{x^2+4}x+C_1$

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