Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- 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
- FOIL Method
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We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\sqrt{x^2+6}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
We can solve the integral $\int\sqrt{u^2-6}du$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting in the original integral, we get
Factor the polynomial $6\sec\left(\theta \right)^2-6$ 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
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta $
The integral of a function times a constant ($\sqrt{6}$) is equal to the constant times the integral of the function
The integral of a function times a constant ($\sqrt{6}$) is equal to the constant times the integral of the function
Simplify $\sqrt{\tan\left(\theta \right)^2}$ 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 $\frac{1}{2}$
Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)$
Divide $2$ by $2$
When multiplying two powers that have the same base ($\sqrt{6}$), you can add the exponents
Simplify $\left(\sqrt{6}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $\frac{1}{2}$ and $n$ equals $2$
Simplify $\sqrt{\tan\left(\theta \right)^2}$ 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 $\frac{1}{2}$
Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)$
Divide $2$ by $2$
Multiply the fraction and term in $\left(\frac{1}{2}\right)\cdot 2$
Divide $2$ by $2$
When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents
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: $\frac{1}{2}x\sqrt{x^2+6}+3\ln\left|\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right|$
Gather the results of all integrals
Combine fractions with common denominator $\sqrt{6}$
The integral $-6\int\sec\left(\theta \right)d\theta$ results in: $-6\ln\left|\frac{u+\sqrt{u^2-6}}{\sqrt{6}}\right|$
Gather the results of all integrals
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