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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 to the problem

$-3\ln\left(\sqrt{x^2+6}+x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$
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Step-by-step Solution

How should I solve this problem?

  • Integrate using basic integrals
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Product of Binomials with Common Term
  • FOIL Method
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Can't find a method? Tell us so we can add it.
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

$6\sqrt{6}\int\frac{\sin\left(\theta \right)^2}{\sqrt{6}\cos\left(\theta \right)^{3}}d\theta$
11

Take the constant $\frac{1}{\sqrt{6}}$ out of the integral

$6\sqrt{6}\cdot \frac{\sqrt{6}}{6}\int\frac{\sin\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}d\theta$
12

Multiply $6\sqrt{6}$ times $\frac{\sqrt{6}}{6}$

$6\int\frac{\sin\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}d\theta$
13

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

$6\int\frac{1-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}d\theta$
14

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

$6\int\left(\frac{1}{\cos\left(\theta \right)^{3}}+\frac{-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}\right)d\theta$
15

Simplify the resulting fractions

$6\int\left(\frac{1}{\cos\left(\theta \right)^{3}}+\frac{-1}{\cos\left(\theta \right)}\right)d\theta$
16

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

$6\int\frac{1}{\cos\left(\theta \right)^{3}}d\theta+6\int\frac{-1}{\cos\left(\theta \right)}d\theta$
17

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

$\frac{1}{2}\sqrt{x^2+6}x+3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
18

Gather the results of all integrals

$3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)+\frac{1}{2}\sqrt{x^2+6}x+6\int\frac{-1}{\cos\left(\theta \right)}d\theta$
19

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

$-6\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
20

Gather the results of all integrals

$3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)+\frac{1}{2}\sqrt{x^2+6}x-6\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
21

Combining like terms $3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$ and $-6\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$

$-3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)+\frac{1}{2}\sqrt{x^2+6}x$
22

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

$L.C.M.=\sqrt{6}$
23

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

$-3\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)+\frac{1}{2}\sqrt{x^2+6}x$
24

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

$-3\ln\left(\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right)+\frac{1}{2}\sqrt{x^2+6}x+C_0$
25

Simplify the expression by applying logarithm properties

$-3\ln\left(\sqrt{x^2+6}+x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$

Final answer to the problem

$-3\ln\left(\sqrt{x^2+6}+x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$

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Plotting: $-3\ln\left(\sqrt{x^2+6}+x\right)+\frac{1}{2}\sqrt{x^2+6}x+C_1$

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
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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