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

The integral of 1/(x^8+x) is 1/7 ln|x^7|-1/7 ln|x^7+1|+C

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integral of 1/(x^8+x)

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

\frac{1}{7} ln x^{7} - \frac{1}{7} ln x^{7} + 1 + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{x ^{7} + 1} - \int - \frac{8 x ^{7} + 1}{x ( x ^{7} + 1 ) ^{2}} d x

= \frac{1}{x ^{7} + 1} - \int - \frac{8 x ^{7} + 1}{x ( x ^{7} + 1 ) ^{2}} d x

\int - \frac{8 x ^{7} + 1}{x ( x ^{7} + 1 ) ^{2}} d x = - \frac{1}{7} ln x^{7} + \frac{1}{7} ln x^{7} + 1 + \frac{1}{x ^{7} + 1}

= \frac{1}{x ^{7} + 1} - (- \frac{1}{7} ln x^{7} + \frac{1}{7} ln x^{7} + 1 + \frac{1}{x ^{7} + 1})

Simplify

= \frac{1}{7} ln x^{7} - \frac{1}{7} ln x^{7} + 1

Add a constant to the solution

= \frac{1}{7} ln x^{7} - \frac{1}{7} ln x^{7} + 1 + C

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What is the integral of 1/(x^8+x) ?
The integral of 1/(x^8+x) is 1/7 ln|x^7|-1/7 ln|x^7+1|+C