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

Step-by-step Solution

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Final Answer

$\frac{\sqrt{2}}{3}\arctan\left(0.707106x\right)-\frac{1}{3}\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right)+\frac{2}{3}\ln\left(x-2\right)+C_0$
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Step-by-step Solution

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

$\frac{x^2}{\left(x^2+2\right)\left(x-2\right)}=\frac{Ax+B}{x^2+2}+\frac{C}{x-2}$

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

$\frac{x^2}{\left(x^2+2\right)\left(x-2\right)}=\frac{Ax+B}{x^2+2}+\frac{C}{x-2}$

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

Final Answer

$\frac{\sqrt{2}}{3}\arctan\left(0.707106x\right)-\frac{1}{3}\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right)+\frac{2}{3}\ln\left(x-2\right)+C_0$

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Function Plot

Plotting: $\frac{\sqrt{2}}{3}\arctan\left(0.707106x\right)-\frac{1}{3}\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right)+\frac{2}{3}\ln\left(x-2\right)+C_0$

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a
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x
y
z
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(◻)
+
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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

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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.

Used Formulas

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