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Find the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$

Step-by-step Solution

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Solution

Final Answer

$-5.999987\ln\left(\sqrt{x^2+6}+x\right)+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$
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Step-by-step Solution

Specify the solving method

1

We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration method of trigonometric substitution using the substitution

$x=\sqrt{6}\tan\left(\theta \right)$
2

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

$dx=\sqrt{6}\sec\left(\theta \right)^2d\theta$
3

Substituting in the original integral, we get

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6\tan\left(\theta \right)^2+6}}d\theta$
4

Factor the polynomial $6\tan\left(\theta \right)^2+6$ by it's greatest common factor (GCF): $6$

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6\left(\tan\left(\theta \right)^2+1\right)}}d\theta$
5

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

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sqrt{\tan\left(\theta \right)^2+1}}d\theta$
6

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

$\int\frac{6\sqrt{6}\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sqrt{\sec\left(\theta \right)^2}}d\theta$
Why is tan(x)^2+1 = sec(x)^2 ?
7

Taking the constant ($6\sqrt{6}$) out of the integral

$6\sqrt{6}\int\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sqrt{\sec\left(\theta \right)^2}}d\theta$
8

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 $\frac{1}{2}$

$6\sqrt{6}\int\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sec\left(\theta \right)}d\theta$
9

Simplify the fraction $\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)^2}{\sqrt{6}\sec\left(\theta \right)}$ by $\sec\left(\theta \right)$

$6\sqrt{6}\int\frac{\tan\left(\theta \right)^2\sec\left(\theta \right)}{\sqrt{6}}d\theta$
10

Simplify the expression inside the integral

$5.999987\int\frac{\sin\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}d\theta$
11

Rewrite the trigonometric expression $\frac{\sin\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}$ inside the integral

$5.999987\int\frac{1-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}d\theta$
12

Expand the fraction $\frac{1-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}$ into $2$ simpler fractions with common denominator $\cos\left(\theta \right)^{3}$

$5.999987\int\left(\frac{1}{\cos\left(\theta \right)^{3}}+\frac{-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}\right)d\theta$
13

Simplify the resulting fractions

$5.999987\int\left(\frac{1}{\cos\left(\theta \right)^{3}}+\frac{-1}{\cos\left(\theta \right)}\right)d\theta$
14

Expand the integral $\int\left(\frac{1}{\cos\left(\theta \right)^{3}}+\frac{-1}{\cos\left(\theta \right)}\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$5.999987\int\frac{1}{\cos\left(\theta \right)^{3}}d\theta+5.999987\int\frac{-1}{\cos\left(\theta \right)}d\theta$
15

The integral $5.999987\int\frac{1}{\cos\left(\theta \right)^{3}}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)$

$\frac{1}{2}\sqrt{x^2+6}x+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)$
16

Gather the results of all integrals

$2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x+5.999987\int\frac{-1}{\cos\left(\theta \right)}d\theta$
17

The integral $5.999987\int\frac{-1}{\cos\left(\theta \right)}d\theta$ results in: $-5.999987\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$

$-5.999987\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
18

Gather the results of all integrals

$2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-5.999987\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
19

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

$L.C.M.=\sqrt{6}$
20

Combine and simplify all terms in the same fraction with common denominator $\sqrt{6}$

$2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-5.999987\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)$
21

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-5.999987\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)+C_0$
22

Simplify the expression by applying logarithm properties

$-5.999987\ln\left(\sqrt{x^2+6}+x\right)+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$

Final Answer

$-5.999987\ln\left(\sqrt{x^2+6}+x\right)+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of ((x^2)/((x^2+6)^0.5))dx using partial fractionsSolve integral of ((x^2)/((x^2+6)^0.5))dx using basic integralsSolve integral of ((x^2)/((x^2+6)^0.5))dx using u-substitutionSolve integral of ((x^2)/((x^2+6)^0.5))dx using integration by partsSolve integral of ((x^2)/((x^2+6)^0.5))dx using trigonometric substitution

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Function Plot

Plotting: $-5.999987\ln\left(\sqrt{x^2+6}+x\right)+2.999994\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$

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asinh
acosh
atanh
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asech
acsch

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