Final answer to the problem
Step-by-step Solution
How should I solve this problem?
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- 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
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We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration method of trigonometric substitution using the substitution
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
Simplifying
The integral of a function times a constant ($6$) is equal to the constant times the integral of the function
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
The integral $6\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{x^{3}}{2\sqrt{x^2+6}}+6\left(\frac{x}{2\sqrt{x^2+6}}\right)+3\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|$
Gather the results of all integrals
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 logarithm properties
