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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{5 ( u - 1 ) u} d u

= \int \frac{1}{5 ( u - 1 ) u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{1}{( u - 1 ) u} d u

Take the partial fraction of

\frac{1}{( u - 1 ) u} : \frac{1}{u - 1} - \frac{1}{u}

= \frac{1}{5} \cdot \int \frac{1}{u - 1} - \frac{1}{u} d u

Apply the Sum Rule:

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

= \frac{1}{5} (\int \frac{1}{u - 1} d u - \int \frac{1}{u} d u)

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{5} (ln ∣ u - 1 ∣ - ln ∣ u ∣)

Substitute back

u = 1 + x^{5}

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{cos ( 5 x )}{5 x} d x

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\int \frac{1}{x} - \frac{5}{x ^{2} + 1} d x

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