Final Answer
Step-by-step Solution
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 greatest common factor (GCF): $6$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
Taking the constant ($6\sqrt{6}$) out of the integral
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 $\frac{1}{2}$
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 $\tan\left(\theta \right)^2\sec\left(\theta \right)$ in terms of sine and cosine
Simplify the fraction $\frac{\frac{\sin\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}}{\sqrt{6}}$
Multiplying the fraction by $\frac{89}{218}$
Taking the constant ($\frac{89}{218}$) out of the integral
Simplify the expression inside the integral
Applying the trigonometric identity: $\sin\left(\theta \right)^2 = 1-\cos\left(\theta \right)^2$
Rewrite the trigonometric expression $\frac{\sin\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}$ inside the integral
Expand the fraction $\frac{1-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}$ into $2$ simpler fractions with common denominator $\cos\left(\theta \right)^{3}$
Simplify the fraction by $\cos\left(\theta \right)$
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Simplify the resulting fractions
Expand the integral $\int\left(\frac{1}{\cos\left(\theta \right)^{3}}-\sec\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Rewrite the trigonometric expression $\frac{1}{\cos\left(\theta \right)^{3}}$ inside the integral
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 $5.999987$ 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 $-5.999987\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$
Simplify the fraction $-5.999987\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$
Express the variable $\theta$ in terms of the original variable $x$
Combining like terms $0.999998\sqrt{x^2+6}x$ 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 $5.999987\int\frac{1}{\cos\left(\theta \right)^{3}}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)$
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 $x$
The integral $5.999987\int-\sec\left(\theta \right)d\theta$ results in: $-5.999987\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
Gather the results of all integrals
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Combine and simplify all terms in the same fraction with common denominator $\sqrt{6}$
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