What is the integral of (8x^3+13x)/(x^4+4x^2+4) ?

The integral of (8x^3+13x)/(x^4+4x^2+4) is 4ln|x^2+2|+3/(2(x^2+2))+C

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integral of (8x^3+13x)/(x^4+4x^2+4)

\int \frac{8 x ^{3} + 13 x}{x ^{4} + 4 x ^{2} + 4} d x

4 ln x^{2} + 2 + \frac{3}{2 ( x ^{2} + 2 )} + C

Solution steps

\int \frac{8 x ^{3} + 13 x}{x ^{4} + 4 x ^{2} + 4} d x

Take the partial fraction of

\frac{8 x ^{3} + 13 x}{x ^{4} + 4 x ^{2} + 4} : \frac{8 x}{x ^{2} + 2} - \frac{3 x}{( x ^{2} + 2 ) ^{2}}

= \int \frac{8 x}{x ^{2} + 2} - \frac{3 x}{( x ^{2} + 2 ) ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{8 x}{x ^{2} + 2} d x - \int \frac{3 x}{( x ^{2} + 2 ) ^{2}} d x

\int \frac{8 x}{x ^{2} + 2} d x = 4 ln x^{2} + 2

\int \frac{3 x}{( x ^{2} + 2 ) ^{2}} d x = - \frac{3}{2 ( x ^{2} + 2 )}

= 4 ln x^{2} + 2 - (- \frac{3}{2 ( x ^{2} + 2 )})

Simplify

= 4 ln x^{2} + 2 + \frac{3}{2 ( x ^{2} + 2 )}

Add a constant to the solution

= 4 ln x^{2} + 2 + \frac{3}{2 ( x ^{2} + 2 )} + C

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What is the integral of (8x^3+13x)/(x^4+4x^2+4) ?
The integral of (8x^3+13x)/(x^4+4x^2+4) is 4ln|x^2+2|+3/(2(x^2+2))+C