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# Find the integral $\int\frac{x^3}{\sqrt{x^2+4}}dx$

## Step-by-step Solution

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###  Videos

$\frac{1}{3}\sqrt{\left(x^2+4\right)^{3}}-4\sqrt{x^2+4}+C_0$
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## Step-by-step Solution

Problem to solve:

$\int\frac{x^3}{\sqrt{x^2+4}}dx$

Specify the solving method

1

We can solve the integral $\int\frac{x^3}{\sqrt{x^2+4}}dx$ by applying integration method of trigonometric substitution using the substitution

$x=2\tan\left(\theta \right)$
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

$dx=2\sec\left(\theta \right)^2d\theta$

Learn how to solve integrals of rational functions problems step by step online.

$x=2\tan\left(\theta \right)$

Learn how to solve integrals of rational functions problems step by step online. Find the integral int((x^3)/((x^2+4)^1/2))dx. We can solve the integral \int\frac{x^3}{\sqrt{x^2+4}}dx by applying integration method of trigonometric substitution using the substitution. 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. Substituting in the original integral, we get. Factor the polynomial 4\tan\left(\theta \right)^2+4 by it's GCF: 4.

$\frac{1}{3}\sqrt{\left(x^2+4\right)^{3}}-4\sqrt{x^2+4}+C_0$

##  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 int((x^3)/((x^2+4)^1/2))dx using partial fractionsSolve int((x^3)/((x^2+4)^1/2))dx using basic integralsSolve int((x^3)/((x^2+4)^1/2))dx using u-substitutionSolve int((x^3)/((x^2+4)^1/2))dx using integration by partsSolve int((x^3)/((x^2+4)^1/2))dx using trigonometric substitution
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### beta Got a different answer? Verify it!

Go!
1
2
3
4
5
6
7
8
9
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

$\int\frac{x^3}{\sqrt{x^2+4}}dx$