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

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##  Final answer to the problem

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

How should I solve this problem?

• Choose an option
• 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
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

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

Simplifying

$\int6\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$
5

The integral of a function times a constant ($6$) is equal to the constant times the integral of the function

$6\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$
6

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

$6\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta$
7

Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$

$6\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$
8

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

$6\int\sec\left(\theta \right)^{3}d\theta-6\int\sec\left(\theta \right)d\theta$
9

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

$\frac{x^{3}}{2\sqrt{x^2+6}}+6\left(\frac{x}{2\sqrt{x^2+6}}\right)+3\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|$
10

Gather the results of all integrals

$3\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|+6\left(\frac{x}{2\sqrt{x^2+6}}\right)+\frac{x^{3}}{2\sqrt{x^2+6}}-6\int\sec\left(\theta \right)d\theta$
11

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

$-6\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|$
12

Gather the results of all integrals

$3\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|+6\left(\frac{x}{2\sqrt{x^2+6}}\right)+\frac{x^{3}}{2\sqrt{x^2+6}}-6\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|$
13

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

$-3\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|+6\left(\frac{x}{2\sqrt{x^2+6}}\right)+\frac{x^{3}}{2\sqrt{x^2+6}}$
14

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

$-3\ln\left|\frac{\sqrt{x^2+6}+x}{\sqrt{6}}\right|+6\left(\frac{x}{2\sqrt{x^2+6}}\right)+\frac{x^{3}}{2\sqrt{x^2+6}}+C_0$
15

Simplify the expression by applying logarithm properties

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

##  Final answer to the problem

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

##  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

SnapXam A2

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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

###  Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).