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 method of trigonometric substitution using the substitution
Differentiate both sides of the equation $x=\sqrt{6}\tan\left(\theta \right)$
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
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 ${f'(x) = sec^2(x)\cdot D_x(x)}$
The derivative of the linear function is equal to $1$
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
Substituting in the original integral, we get
Factor the polynomial $6\tan\left(\theta \right)^2+6$ by it's GCF: $6$
The power of a product is equal to the product of it's factors raised to the same power
Calculate the power $\left(\sqrt{6}\right)^2$
Calculate the power $\left(\sqrt{6}\right)^2$
Calculate the power $\sqrt{6}$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
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 $\frac{1}{2}$
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
Taking the constant ($6\sqrt{6}$) out of the integral
Simplify the fraction $\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sec\left(\theta \right)}$ by $\sec\left(\theta \right)$
Rewrite the fraction $\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)}{\sqrt{6}}$
The integral of a function times a constant ($\frac{\sqrt{6}}{6}$) is equal to the constant times the integral of the function
Multiply $6\sqrt{6}$ times $\frac{\sqrt{6}}{6}$
The integral of a function times a constant ($\frac{\sqrt{6}}{6}$) is equal to the constant times the integral of the function
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 $6$ 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 $x$
Multiplying the fraction by $6\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}\right)$
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 $
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 $x$
Simplifying
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 $-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
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 $x$
Multiplying the fraction by $-3\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}\right)^{2}$
Simplify the fraction by $x^2+6$
Combining like terms $x\sqrt{x^2+6}$ and $-\frac{1}{2}x\sqrt{x^2+6}$
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 $x$
Simplifying
The integral $6\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{1}{2}x\sqrt{x^2+6}-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)+6\ln\left(\frac{\sqrt{6}}{6}\sqrt{x^2+6}+\frac{\sqrt{6}}{6}x\right)$
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 $x$
Simplifying
The integral $6\int-\sec\left(\theta \right)d\theta$ results in: $-6\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)$
Gather the results of all integrals
Combining like terms $-3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)$ and $-6\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$
Multiplying polynomials $-9$ and $-0.89588+\ln\left(\sqrt{x^2+6}+x\right)$
We can combine and rename $8.062918+C_0$ as other constant of integration
Simplify the expression by applying logarithm properties