What is the integral of (x^8)/(1+x^2) ?

The integral of (x^8)/(1+x^2) is (x^7)/7-(x^5)/5+(x^3)/3-x+arctan(x)+C

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integral of (x^8)/(1+x^2)

\int \frac{x ^{8}}{1 + x ^{2}} d x

\frac{x ^{7}}{7} - \frac{x ^{5}}{5} + \frac{x ^{3}}{3} - x + arctan (x) + C

Solution steps

\int \frac{x ^{8}}{1 + x ^{2}} d x

Long division

\frac{x ^{8}}{1 + x ^{2}} : x^{6} - x^{4} + x^{2} - 1 + \frac{1}{x ^{2} + 1}

= \int x^{6} - x^{4} + x^{2} - 1 + \frac{1}{x ^{2} + 1} 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^{6} d x - \int x^{4} d x + \int x^{2} d x - \int 1 d x + \int \frac{1}{x ^{2} + 1} d x

\int x^{6} d x = \frac{x ^{7}}{7}

\int x^{4} d x = \frac{x ^{5}}{5}

\int x^{2} d x = \frac{x ^{3}}{3}

\int 1 d x = x

\int \frac{1}{x ^{2} + 1} d x = arctan (x)

= \frac{x ^{7}}{7} - \frac{x ^{5}}{5} + \frac{x ^{3}}{3} - x + arctan (x)

Add a constant to the solution

= \frac{x ^{7}}{7} - \frac{x ^{5}}{5} + \frac{x ^{3}}{3} - x + arctan (x) + C

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What is the integral of (x^8)/(1+x^2) ?
The integral of (x^8)/(1+x^2) is (x^7)/7-(x^5)/5+(x^3)/3-x+arctan(x)+C