# Step-by-step Solution

## Find the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$

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$\frac{1}{2}x\sqrt{x^2+6}-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|+C_0$

## Step-by-step Solution

Problem to solve:

$\int\frac{x^2}{\sqrt{x^2+6}}dx$

Solving method

1

We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration method of trigonometric substitution using the substitution

$x=\frac{6}{\sqrt{6}}\tan\left(\theta \right)$

Differentiate both sides of the equation $x=\frac{6}{\sqrt{6}}\tan\left(\theta \right)$

$dx=\frac{d}{d\theta}\left(\frac{6}{\sqrt{6}}\tan\left(\theta \right)\right)$

Find the derivative

$\frac{d}{d\theta}\left(\frac{6}{\sqrt{6}}\tan\left(\theta \right)\right)$

The derivative of a function multiplied by a constant ($\frac{6}{\sqrt{6}}$) is equal to the constant times the derivative of the function

$\frac{6}{\sqrt{6}}\frac{d}{d\theta}\left(\tan\left(\theta \right)\right)$

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 ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\frac{6}{\sqrt{6}}\sec\left(\theta \right)^2\frac{d}{d\theta}\left(\theta \right)$

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

$\frac{6}{\sqrt{6}}\sec\left(\theta \right)^2$
2

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

$dx=\frac{6}{\sqrt{6}}\sec\left(\theta \right)^2d\theta$
3

Substituting in the original integral, we get

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

Factor by the greatest common divisor $6$

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

$\int\frac{6}{\sqrt{6}}\left(\frac{\frac{6}{\sqrt{6}}^2\tan\left(\theta \right)^2}{\sqrt{\left(\frac{6}{\sqrt{6}}\tan\left(\theta \right)\right)^2+6}}\right)\sec\left(\theta \right)^2d\theta$

Calculate the power $\frac{6}{\sqrt{6}}^2$

$\int\frac{6}{\sqrt{6}}\left(\frac{6\tan\left(\theta \right)^2}{\sqrt{\left(\frac{6}{\sqrt{6}}\tan\left(\theta \right)\right)^2+6}}\right)\sec\left(\theta \right)^2d\theta$

Multiplying the fraction by $\frac{6}{\sqrt{6}}$

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{\left(\frac{6}{\sqrt{6}}\tan\left(\theta \right)\right)^2+6}}d\theta$

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{\frac{6}{\sqrt{6}}^2\tan\left(\theta \right)^2+6}}d\theta$

Calculate the power $\frac{6}{\sqrt{6}}^2$

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6\tan\left(\theta \right)^2+6}}d\theta$

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

Calculate the power $\sqrt{6}$

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\frac{6}{\sqrt{6}}\sqrt{\tan\left(\theta \right)^2+1}}d\theta$
5

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

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

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

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\frac{6}{\sqrt{6}}\sec\left(\theta \right)}d\theta$
7

Taking the constant ($6\sqrt{6}$) out of the integral

$6\sqrt{6}\int\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\frac{6}{\sqrt{6}}\sec\left(\theta \right)}d\theta$
8

Simplify the fraction by $\sec\left(\theta \right)$

$6\sqrt{6}\int\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)}{\frac{6}{\sqrt{6}}}d\theta$
9

Rewrite the fraction $\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)}{\frac{6}{\sqrt{6}}}$

$6\sqrt{6}\int\frac{\sqrt{6}}{6}\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$

$6\sqrt{6}\cdot \frac{\sqrt{6}}{6}\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$

Multiply $6\sqrt{6}$ times $\frac{\sqrt{6}}{6}$

$6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$
10

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$

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

$6\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta$

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

$6\int\left(\sec\left(\theta \right)\sec\left(\theta \right)^2-\sec\left(\theta \right)\right)d\theta$

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

$6\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$

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

$6\left(\int\sec\left(\theta \right)^{3}d\theta+\int-\sec\left(\theta \right)d\theta\right)$
11

Apply the formula: $\int\sec\left(x\right)\tan\left(x\right)^2dx$$=\int\sec\left(x\right)^3dx-\int\sec\left(x\right)dx, where x=\theta 6\left(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta\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| 6\left(\int\sec\left(\theta \right)^3d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) 6\left(\int\sec\left(\theta \right)^2\sec\left(\theta \right)^{\left(3-2\right)}d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) Subtract the values 3 and -2 6\left(\int\sec\left(\theta \right)^2\sec\left(\theta \right)^{1}d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) Any expression to the power of 1 is equal to that same expression 6\left(\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) 13 Rewrite \sec\left(\theta \right)^3 as the product of two secants 6\left(\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) 14 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) \sec\left(\theta \right)\tan\left(\theta \right) 15 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} 16 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} 17 Solve the integral v=\int\sec\left(\theta \right)^2d\theta 18 The integral of \sec(x)^2 is \tan(x) \tan\left(\theta \right) 19 Now replace the values of u, du and v in the last formula 6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)\tan\left(\theta \right)\tan\left(\theta \right)d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) 20 When multiplying two powers that have the same base (\tan\left(\theta \right)), you can add the exponents 6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right) Applying the trigonometric identity: \tan^2(\theta)=\sec(\theta)^2-1 6\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta Multiplying polynomials \sec\left(\theta \right) and \sec\left(\theta \right)^2-1 6\int\left(\sec\left(\theta \right)\sec\left(\theta \right)^2-\sec\left(\theta \right)\right)d\theta When multiplying exponents with same base you can add the exponents: \sec\left(\theta \right)\sec\left(\theta \right)^2 6\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta The integral of the sum of two or more functions is equal to the sum of their integrals 6\left(\int\sec\left(\theta \right)^{3}d\theta+\int-\sec\left(\theta \right)d\theta\right) Applying the trigonometric identity: \tan^2(\theta)=\sec(\theta)^2-1 -\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta Multiplying polynomials \sec\left(\theta \right) and \sec\left(\theta \right)^2-1 -\int\left(\sec\left(\theta \right)\sec\left(\theta \right)^2-\sec\left(\theta \right)\right)d\theta When multiplying exponents with same base you can add the exponents: \sec\left(\theta \right)\sec\left(\theta \right)^2 -\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta The integral of the sum of two or more functions is equal to the sum of their integrals -\left(\int\sec\left(\theta \right)^{3}d\theta+\int-\sec\left(\theta \right)d\theta\right) 21 Apply the formula: \int\sec\left(x\right)\tan\left(x\right)^2dx$$=\int\sec\left(x\right)^3dx-\int\sec\left(x\right)dx$, where $x=\theta$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta\right)-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right)$
22

Solve the product $-(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta)$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)^3d\theta+\int\sec\left(\theta \right)d\theta-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right)$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)^3d\theta+\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|-\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|\right)$

Subtracting $\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|$ and $\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)^3d\theta\right)$
23

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

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)^3d\theta\right)$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{\left(3-1\right)}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)\right)$

Subtract the values $3$ and $-1$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{\left(3-1\right)}}{2}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)\right)$

Add the values $3$ and $-1$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)\right)$

Subtract the values $3$ and $-1$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{3-2}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)\right)$

Add the values $3$ and $-2$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)\right)$

Divide $1$ by $2$

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)\right)$

Subtract the values $3$ and $-2$

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

Any expression to the power of $1$ is equal to that same expression

$6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)\right)$
24

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

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

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

Multiply $6$ times $-1$

$6\tan\left(\theta \right)\sec\left(\theta \right)-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
25

Solve the product $6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)\right)$

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

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

Multiply $6$ times $-1$

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

$6\tan\left(\theta \right)\sec\left(\theta \right)-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)-6\cdot \frac{1}{2}\int\sec\left(\theta \right)d\theta$

Multiply $-6$ times $\frac{1}{2}$

$6\tan\left(\theta \right)\sec\left(\theta \right)-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)-3\int\sec\left(\theta \right)d\theta$

Multiplying the fraction by $-6$

$6\tan\left(\theta \right)\sec\left(\theta \right)+\frac{-6\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}-3\int\sec\left(\theta \right)d\theta$

Take $\frac{-6}{2}$ out of the fraction

$6\tan\left(\theta \right)\sec\left(\theta \right)-3\sin\left(\theta \right)\sec\left(\theta \right)^{2}-3\int\sec\left(\theta \right)d\theta$
26

Solve the product $-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$

$6\tan\left(\theta \right)\sec\left(\theta \right)-3\sin\left(\theta \right)\sec\left(\theta \right)^{2}-3\int\sec\left(\theta \right)d\theta$

Apply the formula: $\sin\left(x\right)\sec\left(x\right)^n$$=\tan\left(x\right)\sec\left(x\right)^{\left(n-1\right)}$, where $x=\theta$ and $n=2$

$6\tan\left(\theta \right)\sec\left(\theta \right)-3\tan\left(\theta \right)\sec\left(\theta \right)-3\int\sec\left(\theta \right)d\theta$

Combining like terms $6\tan\left(\theta \right)\sec\left(\theta \right)$ and $-3\tan\left(\theta \right)\sec\left(\theta \right)$

$3\tan\left(\theta \right)\sec\left(\theta \right)-3\int\sec\left(\theta \right)d\theta$
27

Simplifying

$3\tan\left(\theta \right)\sec\left(\theta \right)-3\int\sec\left(\theta \right)d\theta$

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

$-3\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|$

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

$-3\ln\left|\frac{\sqrt{x^2+6}}{\frac{6}{\sqrt{6}}}+\frac{x}{\frac{6}{\sqrt{6}}}\right|$

Simplifying

$-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$
28

The integral $-3\int\sec\left(\theta \right)d\theta$ results in: $-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$

$-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$
29

Gather the results of all integrals

$3\tan\left(\theta \right)\sec\left(\theta \right)-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$
30

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

$3\left(\frac{x}{\frac{6}{\sqrt{6}}}\right)\left(\frac{\sqrt{x^2+6}}{\frac{6}{\sqrt{6}}}\right)-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$
31

Multiplying the fraction by $3$

$\frac{3x}{\frac{6}{\sqrt{6}}}\frac{\sqrt{x^2+6}}{\frac{6}{\sqrt{6}}}-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$
32

Take $\frac{3}{\frac{6}{\sqrt{6}}}$ out of the fraction

$\frac{1}{2}x\sqrt{x^2+6}-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|$
33

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

$\frac{1}{2}x\sqrt{x^2+6}-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|+C_0$

$\frac{1}{2}x\sqrt{x^2+6}-3\ln\left|\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right|+C_0$
$\int\frac{x^2}{\sqrt{x^2+6}}dx$