Step-by-step Solution

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 $1$

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

Calculate the power $\left(\sqrt{6}\right)^2$

Multiplying the fraction by $\sqrt{6}$

Calculate the power $\left(\sqrt{6}\right)^2$

Calculate the power $\sqrt{6}$

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

Applying the power of a power property

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

Multiply $6\sqrt{6}$ times $\frac{\sqrt{6}}{6}$

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

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

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 $\frac{3}{\sqrt{6}}$ out of the fraction

Simplifying

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

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

Simplifying

Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$

Multiplying polynomials $-3$ and $-0.89588+\ln\left(\sqrt{x^2+6}+x\right)$

We can combine and rename $2.687639+C_0$ as other constant of integration