Final Answer
Step-by-step Solution
Problem to solve:
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 $x=\frac{6}{\sqrt{6}}\tan\left(\theta \right)$
Find the derivative
The derivative of a function multiplied by a constant ($\frac{6}{\sqrt{6}}$) 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 by the greatest common divisor $6$
Calculate the power $\frac{6}{\sqrt{6}}^2$
Multiplying the fraction by $\frac{6}{\sqrt{6}}$
Calculate the power $\frac{6}{\sqrt{6}}^2$
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: $\tan(x)^2+1=\sec(x)^2$
Taking the constant ($6\sqrt{6}$) out of the integral
Simplify the fraction by $\sec\left(\theta \right)$
Rewrite the fraction $\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)}{\frac{6}{\sqrt{6}}}$
Multiply $6\sqrt{6}$ times $\frac{\sqrt{6}}{6}$
The integral of a constant by a function is equal to the constant multiplied by the integral of the function
Applying the trigonometric identity: $\tan^2(\theta)=\sec(\theta)^2-1$
Multiplying polynomials $\sec\left(\theta \right)$ and $\sec\left(\theta \right)^2-1$
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)\sec\left(\theta \right)^2$
The integral of the sum of two or more functions is equal to the sum of their integrals
Solve the product $6\left(\int\sec\left(\theta \right)^3d\theta-\int\sec\left(\theta \right)d\theta\right)$
Apply the formula: $\int\sec\left(x\right)\tan\left(x\right)^2dx$$=\int\sec\left(x\right)^3dx-\int\sec\left(x\right)dx$, where $x=\theta $
Rewrite $\sec\left(\theta \right)^3$ as the product of two secants
We can solve the integral $\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The integral of $\sec(x)^2$ is $\tan(x)$
Now replace the values of $u$, $du$ and $v$ in the last formula
Solve the product $6\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$
Apply the formula: $\int\sec\left(x\right)\tan\left(x\right)^2dx$$=\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, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
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 $-6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
Simplifying
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Combining like terms $-3\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$ and $6\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$
Express the variable $\theta$ in terms of the original variable $x$
Multiplying the fraction by $3$
Take $\frac{3}{\frac{6}{\sqrt{6}}}$ out of the fraction
Simplifying
The integral $6\int\sec\left(\theta \right)^3d\theta$ results in: $\frac{1}{2}x\sqrt{x^2+6}+3\ln\left(\frac{\sqrt{6}}{6}\left(\sqrt{x^2+6}+x\right)\right)$
Gather the results of all integrals
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 $C$
Simplify the expression by applying logarithm properties