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

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

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

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

\frac{x ^{4}}{4} - 2 x^{2} + 8 ln x^{2} + 4 + C

Solution steps

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

Long division

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

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

\int x^{3} d x = \frac{x ^{4}}{4}

\int 4 x d x = 2 x^{2}

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

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

Add a constant to the solution

= \frac{x ^{4}}{4} - 2 x^{2} + 8 ln x^{2} + 4 + C

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