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

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 $1$

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

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

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

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

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

Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ 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 $x$

Combining like terms $\sqrt{x^2+6}x$ 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

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