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Find the integral $\int\frac{x^3}{1+x^4}dx$

Step-by-step Solution

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Solution

Formulas

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Final Answer

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

Problem to solve:

$\int\frac{x^3}{1+x^4}dx$

Specify the solving method

1

Rewrite the expression $\frac{x^3}{1+x^4}$ inside the integral in factored form

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

Learn how to solve integrals by partial fraction expansion problems step by step online.

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

Unlock the first 3 steps of this solution!

Learn how to solve integrals by partial fraction expansion problems step by step online. Find the integral int((x^3)/(1+x^4))dx. Rewrite the expression \frac{x^3}{1+x^4} inside the integral in factored form. Rewrite the fraction \frac{x^3}{\left(x^2-\sqrt{2}x+1\right)\left(x^2+\sqrt{2}x+1\right)} in 2 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D. The first step is to multiply both sides of the equation from the previous step by \left(x^2-\sqrt{2}x+1\right)\left(x^2+\sqrt{2}x+1\right). Multiplying polynomials.

Final Answer

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

Explore different ways to solve this problem

Integrals by Partial Fraction expansionBasic IntegralsIntegration by SubstitutionIntegration by PartsIntegration by Trigonometric Substitution
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Go!
1
2
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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
D□x
∫
∫◻
|◻|
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

Useful tips on how to improve your answer:

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$\int\frac{x^3}{1+x^4}dx$

Main topic:

Integrals by Partial Fraction expansion

Used formulas:

5. See formulas

Time to solve it:

~ 1.04 s

Related topics:

Integrals of Rational FunctionsIntegral CalculusCalculusIntegrals by Partial Fraction expansionMatricesGaussian EliminationIntegration by Trigonometric SubstitutionIntegration TechniquesFactorizationIntegrals of polynomial functions

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