abreviatsjw

2022-01-17

Evaluate the following integral.
$\int \frac{2{x}^{3}-11{x}^{2}+2x-16}{\left({x}^{2}-4\right)\left({x}^{2}+1\right)}dx$

macalpinee3

Expert

Step 1
Evaluate
$\int \frac{2{x}^{3}-11{x}^{2}+2x-16}{\left({x}^{2}-4\right)\left({x}^{2}+1\right)}dx$
Step 2
Simplify the denominator by factorization.
$\int \frac{2{x}^{3}-11{x}^{2}+2x-16}{\left(x+2\right)\left(x-2\right)\left({x}^{2}+1\right)}dx$
Step 3
Now simplify by partial fraction decomposition we get
$\int \left(\frac{1}{{x}^{2}+1}+\frac{4}{x+2}-\frac{2}{x-2}\right)dx$
$=\int \frac{1}{{x}^{2}+1}dx+\int \frac{4}{x+2}dx-\int \frac{2}{x-2}dx$
Step 4
Now integrating we get
$\int \frac{1}{{x}^{2}+1}dx+4\int \frac{1}{x+2}dx-2\int \frac{1}{x-2}dx$
$={\mathrm{tan}}^{-1}\left(x\right)+4\mathrm{ln}\left(x+2\right)-2\mathrm{ln}\left(x-2\right)+c$
Step 5
$\int \frac{2{x}^{3}-11{x}^{2}+2x-16}{\left({x}^{2}-4\right)\left({x}^{2}+1\right)}dx={\mathrm{tan}}^{-1}\left(x\right)+4\mathrm{ln}\left(x+2\right)-2\mathrm{ln}\left(x-2\right)+c$