What is the integral of 5/(x(x^5+4)) ?

The integral of 5/(x(x^5+4)) is 1/4 ln|x^5|-1/4 ln|x^5+4|+C

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integral of 5/(x(x^5+4))

\int \frac{5}{x ( x ^{5} + 4 )} d x

\frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4 + C

Solution steps

\int \frac{5}{x ( x ^{5} + 4 )} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x ( x ^{5} + 4 )} d x

Apply u-substitution

: \int \frac{1}{5 u ( u + 4 )} d u

= 5 \cdot \int \frac{1}{5 u ( u + 4 )} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{u ( u + 4 )} d u

Take the partial fraction of

\frac{1}{u ( u + 4 )} : \frac{1}{4 u} - \frac{1}{4 ( u + 4 )}

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{4 u} - \frac{1}{4 ( u + 4 )} 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

= 5 \cdot \frac{1}{5} (\int \frac{1}{4 u} d u - \int \frac{1}{4 ( u + 4 )} d u)

\int \frac{1}{4 u} d u = \frac{1}{4} ln ∣ u ∣

\int \frac{1}{4 ( u + 4 )} d u = \frac{1}{4} ln ∣ u + 4 ∣

= 5 \cdot \frac{1}{5} (\frac{1}{4} ln ∣ u ∣ - \frac{1}{4} ln ∣ u + 4 ∣)

Substitute back

u = x^{5}

= 5 \cdot \frac{1}{5} (\frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4)

Simplify

5 \cdot \frac{1}{5} (\frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4) : \frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4

= \frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4

Add a constant to the solution

= \frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4 + C

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What is the integral of 5/(x(x^5+4)) ?
The integral of 5/(x(x^5+4)) is 1/4 ln|x^5|-1/4 ln|x^5+4|+C