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

The integral of (x^4)/(5x^5+4) is 1/25 ln|5x^5+4|+C

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

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

\frac{1}{25} ln 5 x^{5} + 4 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{25 u} d u

= \int \frac{1}{25 u} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{25} ln ∣ u ∣

Substitute back

u = 5 x^{5} + 4

= \frac{1}{25} ln 5 x^{5} + 4

Add a constant to the solution

= \frac{1}{25} ln 5 x^{5} + 4 + C

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\frac{d}{d x} (e^{\frac{l n ( x )}{x}})

\frac{d y}{d x} = e^{9 x - 8 y}

d y + 0.15 (y - 30) d x = 0

What is the integral of (x^4)/(5x^5+4) ?
The integral of (x^4)/(5x^5+4) is 1/25 ln|5x^5+4|+C