What is the integral of (x^5)/((x^2+4)^2) ?

The integral of (x^5)/((x^2+4)^2) is (x^2)/2-1/4 x^3sin(2arctan(x/2))+x^2-4ln|x^2+4|+4+8/(x^2+4)+C

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integral of (x^5)/((x^2+4)^2)

\int \frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} d x

\frac{x ^{2}}{2} - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4} + C

Solution steps

\int \frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} d x

Long division

\frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} : x + \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16}

= \int x + \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} 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 x d x + \int \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} d x

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

\int \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} d x = - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4}

= \frac{x ^{2}}{2} - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4}

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4} + C

\frac{d}{d x} (6 x^{3} - 4 x^{2} + 5 x)

f (x) = \frac{9 x}{x + 4}, at x = 8

\int - \frac{1}{4} sin (2 x) d x

f (x) = \frac{6 x}{x + 6}

\frac{d}{d x} (ln (3 x^{2} + 5))

What is the integral of (x^5)/((x^2+4)^2) ?
The integral of (x^5)/((x^2+4)^2) is (x^2)/2-1/4 x^3sin(2arctan(x/2))+x^2-4ln|x^2+4|+4+8/(x^2+4)+C