What is the integral of (2x^5-5x)/((x^2+2)^2) ?

The integral of (2x^5-5x)/((x^2+2)^2) is x^2-4ln|x^2+2|-3/(2(x^2+2))+C

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integral of (2x^5-5x)/((x^2+2)^2)

\int \frac{2 x ^{5} - 5 x}{( x ^{2} + 2 ) ^{2}} d x

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

Solution steps

\int \frac{2 x ^{5} - 5 x}{( x ^{2} + 2 ) ^{2}} d x

Take the partial fraction of

\frac{2 x ^{5} - 5 x}{( x ^{2} + 2 ) ^{2}} : 2 x - \frac{8 x}{x ^{2} + 2} + \frac{3 x}{( x ^{2} + 2 ) ^{2}}

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

\int 2 x d x = x^{2}

\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 )}

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

Add a constant to the solution

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

What is the integral of (2x^5-5x)/((x^2+2)^2) ?
The integral of (2x^5-5x)/((x^2+2)^2) is x^2-4ln|x^2+2|-3/(2(x^2+2))+C