Final Answer
Step-by-step Solution
Problem to solve:
Specify the solving method
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
Differentiate both sides of the equation $u=\sqrt{x^2+6}$
Find the derivative
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of the constant function ($6$) is equal to zero
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
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
Isolate $dx$ in the previous equation
Removing the variable's exponent raising both sides of the equation to the power of $2$
Divide $1$ by $\frac{1}{2}$
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 $2$
Multiply $\frac{1}{2}$ times $2$
Multiply $\frac{1}{2}$ times $2$
Divide $1$ by $\frac{1}{2}$
We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $6$ from both sides of the equation
Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{2}$
Removing the variable's exponent raising both sides of the equation to the power of $2$
Divide $1$ by $\frac{1}{2}$
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 $2$
Multiply $\frac{1}{2}$ times $2$
Multiply $\frac{1}{2}$ times $2$
Divide $1$ by $\frac{1}{2}$
Divide $1$ by $2$
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 $2$
Multiply $\frac{1}{2}$ times $2$
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 $\frac{1}{2}$
Multiply $2$ times $\frac{1}{2}$
Multiply $2$ times $\frac{1}{2}$
Divide $1$ by $2$
Rewriting $x$ in terms of $u$
Divide fractions $\frac{\frac{x^2}{u}}{\left(x^2+6\right)^{-\frac{1}{2}}x}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Simplify the fraction $\frac{x^2}{u\left(x^2+6\right)^{-\frac{1}{2}}x}$ by $x$
Rewriting $x$ in terms of $\sqrt{u^{2}-6}$
Cancel exponents $\frac{1}{2}$ and $2$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiplying the fraction by $u$
Any expression divided by one ($1$) is equal to that same expression
Substituting $u$, $dx$ and $x$ in the integral and simplify
We can solve the integral $\int\sqrt{u^{2}-6}du$ by applying integration method of trigonometric substitution using the substitution
Differentiate both sides of the equation $u=\sqrt{6}\sec\left(\theta \right)$
Find the derivative
The derivative of a function multiplied by a constant ($\sqrt{6}$) is equal to the constant times the derivative of the function
Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
The derivative of the linear function is equal to $1$
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
Substituting in the original integral, we get
Factor the polynomial $6\sec\left(\theta \right)^{2}-6$ by it's greatest common factor (GCF): $6$
The power of a product is equal to the product of it's factors raised to the same power
Apply the trigonometric identity: $\sec\left(x\right)^2-1$$=\tan\left(x\right)^2$, where $x=\theta $
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
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}$
When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents
Applying the trigonometric identity: $\tan^2(\theta)=\sec(\theta)^2-1$
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
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)^2\sec\left(\theta \right)$
Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
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
Rewrite $\sec\left(\theta \right)^{3}$ as the product of two secants
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
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The integral of $\sec(x)^2$ is $\tan(x)$
Now replace the values of $u$, $du$ and $v$ in the last formula
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)$
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$
Subtract the values $6$ and $-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}$
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 $
Solve the product $-6\left(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta\right)$
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
Simplify the expression inside the integral
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$
Subtract the values $6$ and $-6$
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 $\frac{1}{2}$
Simplify the integral $\int\sec\left(\theta \right)^3d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
Solve the product $-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$
Solve the product $\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)$
Simplify the fraction $-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$
Subtract the values $6$ and $-6$
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 $\frac{1}{2}$
Combining like terms $x\sqrt{x^2+6}$ and $-\frac{1}{2}\sqrt{x^2+6}x$
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
Simplify the expression inside the integral
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{x^2+6}$
Subtract the values $6$ and $-6$
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 $\frac{1}{2}$
The integral $\frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$
Gather the results of all integrals
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
Simplify the expression inside the integral
The integral $\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$ results in: $-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{\sqrt{6}}{6}\left(u+\sqrt{u^{2}-6}\right)\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Simplify the expression by applying logarithm properties