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

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Solution

Final answer to the problem

$\frac{1}{10}\ln\left|\sqrt{x^2+2}\right|+\frac{1}{-2\left(x^2+2\right)}+\frac{-3x}{32\left(x^2+2\right)}+\frac{-3\sqrt{2}\arctan\left(\frac{x}{\sqrt{2}}\right)}{64}+\frac{-x}{8\left(x^2+2\right)^{2}}+C_1$
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Step-by-step Solution

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  • Integrate by partial fractions
  • Integrate by substitution
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  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
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  • Integrate using basic integrals
  • Product of Binomials with Common Term
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1

Rewrite the fraction $\frac{x^3+x-1}{\left(x^2+2\right)^3}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{\frac{1}{10}x}{x^2+2}+\frac{x}{\left(x^2+2\right)^{2}}+\frac{-1}{\left(x^2+2\right)^{3}}$

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$\frac{\frac{1}{10}x}{x^2+2}+\frac{x}{\left(x^2+2\right)^{2}}+\frac{-1}{\left(x^2+2\right)^{3}}$

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Unlock the first 3 steps of this solution

Learn how to solve problems step by step online. Find the integral int((x^3+x+-1)/((x^2+2)^3))dx. Rewrite the fraction \frac{x^3+x-1}{\left(x^2+2\right)^3} in 3 simpler fractions using partial fraction decomposition. Simplify the expression. The integral \frac{1}{10}\int\frac{x}{x^2+2}dx results in: -\frac{1}{10}\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right). The integral \int\frac{x}{\left(x^2+2\right)^{2}}dx results in: \frac{1}{-2\left(x^2+2\right)}.

Final answer to the problem

$\frac{1}{10}\ln\left|\sqrt{x^2+2}\right|+\frac{1}{-2\left(x^2+2\right)}+\frac{-3x}{32\left(x^2+2\right)}+\frac{-3\sqrt{2}\arctan\left(\frac{x}{\sqrt{2}}\right)}{64}+\frac{-x}{8\left(x^2+2\right)^{2}}+C_1$

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Function Plot

Plotting: $\frac{1}{10}\ln\left(\sqrt{x^2+2}\right)+\frac{1}{-2\left(x^2+2\right)}+\frac{-3x}{32\left(x^2+2\right)}+\frac{-3\sqrt{2}\arctan\left(\frac{x}{\sqrt{2}}\right)}{64}+\frac{-x}{8\left(x^2+2\right)^{2}}+C_1$

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0
a
b
c
d
f
g
m
n
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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

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