Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Choose an option
- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- Load more...
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 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
The power of a product is equal to the product of it's factors raised to the same power
Multiplying the fraction by $\sqrt{6}\sec\left(\theta \right)^2$
The power of a product is equal to the product of it's factors raised to the same power
Simplify $6\tan\left(\theta \right)^2+6$ into secant function
The power of a product is equal to the product of it's factors raised to the same power
Simplify the fraction $\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sec\left(\theta \right)}$ by $\sqrt{6}$
Substituting in the original integral, we get
Simplify the fraction $\frac{6\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sec\left(\theta \right)}$ by $\sec\left(\theta \right)$
Simplifying
The integral of a function times a constant ($6$) is equal to the constant times the integral of the function
Applying the trigonometric identity: $\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2-1$
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
Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
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
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 $6\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
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 $x$
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 $6\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{1}{2}x\sqrt{x^2+6}+3\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)$
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 $x$
The integral $-6\int\sec\left(\theta \right)d\theta$ results in: $-6\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)$
Gather the results of all integrals
Combining like terms $3\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)$ and $-6\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)$
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 the property of the logarithm of a quotient
