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

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##  Final Answer

$-3\ln\left(\sqrt{x^2+6}+x\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)+C_1$
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##  Step-by-step Solution 

Problem to solve:

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

Specify the solving method

We could not solve this problem by using the method: Integrals by Partial Fraction Expansion

1

We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\sqrt{x^2+6}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=\sqrt{x^2+6}$

Differentiate both sides of the equation $u=\sqrt{x^2+6}$

$du=\frac{d}{dx}\left(\sqrt{x^2+6}\right)$

Find the derivative

$\frac{d}{dx}\left(\sqrt{x^2+6}\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

The derivative of a sum of two or more functions is the sum of the derivatives of each function

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

The derivative of the constant function ($6$) is equal to zero

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

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

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

$du=\left(x^2+6\right)^{-\frac{1}{2}}xdx$
3

Isolate $dx$ in the previous equation

$\frac{du}{\left(x^2+6\right)^{-\frac{1}{2}}x}=dx$

Removing the variable's exponent raising both sides of the equation to the power of $2$

$\left(\sqrt{x^2+6}\right)^{\frac{1}{\frac{1}{2}}}=u^{\frac{1}{\frac{1}{2}}}$

Divide $1$ by $\frac{1}{2}$

$\left(\sqrt{x^2+6}\right)^{2}=u^{\frac{1}{\frac{1}{2}}}$

Simplify $\left(\sqrt{x^2+6}\right)^{2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $\frac{1}{2}$ and $n$ equals $2$

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

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

$x^2+6$

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

$x^2+6=u^{\frac{1}{\frac{1}{2}}}$

Divide $1$ by $\frac{1}{2}$

$x^2+6=u^{2}$

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $6$ from both sides of the equation

$x^2=u^{2}-6$

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{2}$

$\left(x^2\right)^{\frac{1}{2}}=\left(u^{2}-6\right)^{\frac{1}{2}}$

Removing the variable's exponent raising both sides of the equation to the power of $2$

$\left(\sqrt{x^2+6}\right)^{\frac{1}{\frac{1}{2}}}=u^{\frac{1}{\frac{1}{2}}}$

Divide $1$ by $\frac{1}{2}$

$\left(\sqrt{x^2+6}\right)^{2}=u^{\frac{1}{\frac{1}{2}}}$

Simplify $\left(\sqrt{x^2+6}\right)^{2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $\frac{1}{2}$ and $n$ equals $2$

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

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

$x^2+6$

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

$x^2+6=u^{\frac{1}{\frac{1}{2}}}$

Divide $1$ by $\frac{1}{2}$

$x^2+6=u^{2}$

Divide $1$ by $2$

$\sqrt{x^2}=\left(u^{2}-6\right)^{\frac{1}{2}}$

Simplify $\left(\sqrt{x^2+6}\right)^{2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $\frac{1}{2}$ and $n$ equals $2$

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

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

$x^2+6$

Simplify $\sqrt{x^2}$ 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 $\frac{1}{2}$

$x^{2\frac{1}{2}}$

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

$x$

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

$x=\left(u^{2}-6\right)^{\frac{1}{2}}$

Divide $1$ by $2$

$x=\sqrt{u^{2}-6}$
4

Rewriting $x$ in terms of $u$

$x=\sqrt{u^{2}-6}$

Divide fractions $\frac{\frac{x^2}{u}}{\left(x^2+6\right)^{-\frac{1}{2}}x}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\int\frac{x^2}{u\left(x^2+6\right)^{-\frac{1}{2}}x}du$

Simplify the fraction $\frac{x^2}{u\left(x^2+6\right)^{-\frac{1}{2}}x}$ by $x$

$\int\frac{x}{u\left(x^2+6\right)^{-\frac{1}{2}}}du$

Rewriting $x$ in terms of $\sqrt{u^{2}-6}$

$\int\frac{\sqrt{u^{2}-6}}{u\left(\left(\sqrt{u^{2}-6}\right)^2+6\right)^{-\frac{1}{2}}}du$

Cancel exponents $\frac{1}{2}$ and $2$

$\int\frac{\sqrt{u^{2}-6}}{u\left(u^{2}-6+6\right)^{-\frac{1}{2}}}du$

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

$\int\frac{\sqrt{u^{2}-6}}{u\frac{1}{\sqrt{u^{2}-6+6}}}du$

Multiplying the fraction by $u$

$\int\frac{\sqrt{u^{2}-6}}{1}du$

Any expression divided by one ($1$) is equal to that same expression

$\int\sqrt{u^{2}-6}du$
5

Substituting $u$, $dx$ and $x$ in the integral and simplify

$\int\sqrt{u^{2}-6}du$
6

We can solve the integral $\int\sqrt{u^{2}-6}du$ by applying integration method of trigonometric substitution using the substitution

$u=\sqrt{6}\sec\left(\theta \right)$

Differentiate both sides of the equation $u=\sqrt{6}\sec\left(\theta \right)$

$du=\frac{d}{d\theta}\left(\sqrt{6}\sec\left(\theta \right)\right)$

Find the derivative

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

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

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

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

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

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

$\sqrt{6}\sec\left(\theta \right)\tan\left(\theta \right)$
7

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

$du=\sqrt{6}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
8

Substituting in the original integral, we get

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

Factor the polynomial $6\sec\left(\theta \right)^{2}-6$ by it's greatest common factor (GCF): $6$

$\int\sqrt{6}\sqrt{6\left(\sec\left(\theta \right)^2-1\right)}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
10

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

$\int\frac{6}{\log^{3}\left(10\right)}\sqrt{\sec\left(\theta \right)^2-1}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
11

Apply the trigonometric identity: $\sec\left(x\right)^2-1$$=\tan\left(x\right)^2, where x=\theta \int\frac{6}{\log^{3}\left(10\right)}\sqrt{\tan\left(\theta \right)^2}\sec\left(\theta \right)\tan\left(\theta \right)d\theta 12 The integral of a function times a constant (\frac{6}{\log^{3}\left(10\right)}) is equal to the constant times the integral of the function \frac{6}{\log^{3}\left(10\right)}\int\sqrt{\tan\left(\theta \right)^2}\sec\left(\theta \right)\tan\left(\theta \right)d\theta 13 Simplify \sqrt{\tan\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} \frac{6}{\log^{3}\left(10\right)}\int\tan\left(\theta \right)\sec\left(\theta \right)\tan\left(\theta \right)d\theta 14 When multiplying two powers that have the same base (\tan\left(\theta \right)), you can add the exponents \frac{6}{\log^{3}\left(10\right)}\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta Applying the trigonometric identity: \tan^2(\theta)=\sec(\theta)^2-1 \frac{6}{\log^{3}\left(10\right)}\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta 15 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 \frac{6}{\log^{3}\left(10\right)}\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) \frac{6}{\log^{3}\left(10\right)}\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta 16 Multiply the single term \sec\left(\theta \right) by each term of the polynomial \left(\sec\left(\theta \right)^2-1\right) \frac{6}{\log^{3}\left(10\right)}\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta 17 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 \frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^{3}d\theta+\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta Rewrite \sec\left(\theta \right)^{3} as the product of two secants \frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta 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 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} 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} Solve the integral v=\int\sec\left(\theta \right)^2d\theta The integral of \sec(x)^2 is \tan(x) \tan\left(\theta \right) Now replace the values of u, du and v in the last formula \frac{6}{\log^{3}\left(10\right)}\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right) Multiply the single term \frac{6}{\log^{3}\left(10\right)} 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) \frac{6}{\log^{3}\left(10\right)}\tan\left(\theta \right)\sec\left(\theta \right)-6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta Express the variable \theta in terms of the original variable x \sqrt{u^{2}-6}u-6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta Replace u with the value that we assigned to it in the beginning: \sqrt{x^2+6} \sqrt{x^2+6-6}\sqrt{x^2+6}-6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta Subtract the values 6 and -6 \sqrt{x^2}\sqrt{x^2+6}-6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta Simplify \sqrt{x^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} x\sqrt{x^2+6}-6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta 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 $x=\theta$

$x\sqrt{x^2+6}-6\left(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta\right)$

Solve the product $-6\left(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta\right)$

$x\sqrt{x^2+6}-6\int\sec\left(\theta \right)^3d\theta+6\int\sec\left(\theta \right)d\theta$

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

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

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

$x\sqrt{x^2+6}-6\int\sec\left(\theta \right)^3d\theta+6\ln\left(\frac{u}{\sqrt{6}}+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)$

Simplify the expression inside the integral

$x\sqrt{x^2+6}-6\int\sec\left(\theta \right)^3d\theta+6\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)$

Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$

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

Subtract the values $6$ and $-6$

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

Simplify $\sqrt{x^2}$ 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 $\frac{1}{2}$

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

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

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

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

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

Solve the product $\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)$

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

Simplify the fraction $-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$

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

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

$x\sqrt{x^2+6}-\frac{1}{2}u\sqrt{u^{2}-6}-3\int\sec\left(\theta \right)d\theta+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$

Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$

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

Subtract the values $6$ and $-6$

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

Simplify $\sqrt{x^2}$ 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 $\frac{1}{2}$

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

Combining like terms $x\sqrt{x^2+6}$ and $-\frac{1}{2}\sqrt{x^2+6}x$

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

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

$\frac{1}{2}\sqrt{x^2+6}x-3\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$

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

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

Simplify the expression inside the integral

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

Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$

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

Subtract the values $6$ and $-6$

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

Simplify $\sqrt{x^2}$ 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 $\frac{1}{2}$

$\frac{1}{2}\sqrt{x^2+6}x-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$
18

The integral $\frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$

$\frac{1}{2}\sqrt{x^2+6}x-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$
19

Gather the results of all integrals

$6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+\frac{1}{2}\sqrt{x^2+6}x+\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$

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

$-\frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)d\theta$

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

$-\frac{6}{\log^{3}\left(10\right)}\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$

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

$-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{u}{\sqrt{6}}+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)$

Simplify the expression inside the integral

$-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)$
20

The integral $\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$ results in: $-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)$

$-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)$
21

Gather the results of all integrals

$6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)$
22

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

$6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)+C_0$
23

Simplify the expression by applying logarithm properties

$-3\ln\left(\sqrt{x^2+6}+x\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)+C_1$

##  Final Answer

$-3\ln\left(\sqrt{x^2+6}+x\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)+C_1$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of ((x^2)/((x^2+6)^0.5))dx using basic integralsSolve integral of ((x^2)/((x^2+6)^0.5))dx using u-substitutionSolve integral of ((x^2)/((x^2+6)^0.5))dx using integration by partsSolve integral of ((x^2)/((x^2+6)^0.5))dx using trigonometric substitution

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