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

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##  Final answer to the problem

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

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

$\int\frac{1}{\left(x^2-\sqrt[4]{4}x+1\right)\left(x^2+\sqrt[4]{4}x+1\right)}dx$

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

$\int\frac{1}{\left(x^2-\sqrt[4]{4}x+1\right)\left(x^2+\sqrt[4]{4}x+1\right)}dx$

Learn how to solve integrals by partial fraction expansion problems step by step online. Find the integral int(1/(x^4+1))dx. Rewrite the expression \frac{1}{x^4+1} inside the integral in factored form. Rewrite the fraction \frac{1}{\left(x^2-\sqrt[4]{4}x+1\right)\left(x^2+\sqrt[4]{4}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[4]{4}x+1\right)\left(x^2+\sqrt[4]{4}x+1\right). Multiplying polynomials.

##  Final answer to the problem

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

##  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

SnapXam A2

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a
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x
y
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◻/◻
/
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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 by Partial Fraction Expansion

The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.