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

Step-by-step Solution

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Solution

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Final Answer

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

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

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Unlock the first 2 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

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

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)/(1+x^4))dx using partial fractionsSolve int((x^3)/(1+x^4))dx using basic integralsSolve int((x^3)/(1+x^4))dx using u-substitutionSolve int((x^3)/(1+x^4))dx using integration by partsSolve int((x^3)/(1+x^4))dx using trigonometric substitution

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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
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

How to improve your answer:

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Main topic:

Integrals by Partial Fraction Expansion

Used formulas:

3. See formulas

Time to solve it:

~ 0.55 s

Related topics:

Integrals of Rational FunctionsIntegral CalculusCalculusIndefinite IntegralsIntegralIntegrals by Partial Fraction ExpansionMatricesGaussian Elimination

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