What is the integral of (1-x^7)/(x(1+x^7)) ?

The integral of (1-x^7)/(x(1+x^7)) is 1/7 ln|x^7|-2/7 ln|1+x^7|+C

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integral of (1-x^7)/(x(1+x^7))

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

\frac{1}{7} ln x^{7} - \frac{2}{7} ln 1 + x^{7} + C

Solution steps

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

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

\frac{1}{7} (ln x^{7} - ln 1 + x^{7}) - \frac{1}{7} ln 1 + x^{7} : \frac{1}{7} ln x^{7} - \frac{2}{7} ln 1 + x^{7}

= \frac{1}{7} ln x^{7} - \frac{2}{7} ln 1 + x^{7}

Add a constant to the solution

= \frac{1}{7} ln x^{7} - \frac{2}{7} ln 1 + x^{7} + C

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What is the integral of (1-x^7)/(x(1+x^7)) ?
The integral of (1-x^7)/(x(1+x^7)) is 1/7 ln|x^7|-2/7 ln|1+x^7|+C