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Step-by-step Solution

Integral of (x^2+3)/((x+1)(x^2+1))

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Answer

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

Step-by-step explanation

Problem to solve:

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

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

$\frac{x^2+3}{\left(x+1\right)\left(x^2+1\right)}=\frac{A}{x+1}+\frac{Bx+C}{x^2+1}$
2

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x+1\right)\left(x^2+1\right)$

$x^2+3=\left(x+1\right)\left(x^2+1\right)\left(\frac{A}{x+1}+\frac{Bx+C}{x^2+1}\right)$

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Answer

$2\ln\left|x+1\right|-\frac{3}{2}\ln\left|x^2+1\right|+3arctan\left(x\right)+C_0$
$\int\frac{x^2+3}{\left(x+1\right)\left(x^2+1\right)}dx$

Main topic:

Integrals by partial fraction expansion

Used formulas:

6. See formulas

Time to solve it:

~ 1.03 seconds

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