What is the integral of (1-x^5)/(x(1+x^5)) ?

The integral of (1-x^5)/(x(1+x^5)) is 1/5 ln|x^5|-2/5 ln|1+x^5|+C

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integral of (1-x^5)/(x(1+x^5))

\int \frac{1 - x ^{5}}{x ( 1 + x ^{5} )} d x

\frac{1}{5} ln x^{5} - \frac{2}{5} ln 1 + x^{5} + C

Solution steps

\int \frac{1 - x ^{5}}{x ( 1 + x ^{5} )} d x

Expand

\frac{1 - x ^{5}}{x ( 1 + x ^{5} )} : \frac{1}{x ( 1 + x ^{5} )} - \frac{x ^{5}}{x ( 1 + x ^{5} )}

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 ^{5} )} d x - \int \frac{x ^{5}}{x ( 1 + x ^{5} )} d x

\int \frac{1}{x ( 1 + x ^{5} )} d x = \frac{1}{5} (ln x^{5} - ln 1 + x^{5})

\int \frac{x ^{5}}{x ( 1 + x ^{5} )} d x = \frac{1}{5} ln 1 + x^{5}

= \frac{1}{5} (ln x^{5} - ln 1 + x^{5}) - \frac{1}{5} ln 1 + x^{5}

Simplify

\frac{1}{5} (ln x^{5} - ln 1 + x^{5}) - \frac{1}{5} ln 1 + x^{5} : \frac{1}{5} ln x^{5} - \frac{2}{5} ln 1 + x^{5}

= \frac{1}{5} ln x^{5} - \frac{2}{5} ln 1 + x^{5}

Add a constant to the solution

= \frac{1}{5} ln x^{5} - \frac{2}{5} ln 1 + x^{5} + C

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