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

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

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

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

4 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2}) - \frac{13}{2 ( x ^{2} + 2 )} + C

Solution steps

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

Expand

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

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 ^{3}}{( x ^{2} + 2 ) ^{2}} d x + \int \frac{13 x}{( x ^{2} + 2 ) ^{2}} d x

\int \frac{8 x ^{3}}{( x ^{2} + 2 ) ^{2}} d x = 4 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2})

\int \frac{13 x}{( x ^{2} + 2 ) ^{2}} d x = - \frac{13}{2 ( x ^{2} + 2 )}

= 4 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2}) - \frac{13}{2 ( x ^{2} + 2 )}

Add a constant to the solution

= 4 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2}) - \frac{13}{2 ( x ^{2} + 2 )} + C

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