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

Step-by-step Solution

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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

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

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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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x
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.
(◻)
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/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Equations

In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions.

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