What is the integral of (x^4-x^3)/(4x^5-5x^4) ?

The integral of (x^4-x^3)/(4x^5-5x^4) is 1/20 ln|4x^5-5x^4|+C

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integral of (x^4-x^3)/(4x^5-5x^4)

\int \frac{x ^{4} - x ^{3}}{4 x ^{5} - 5 x ^{4}} d x

\frac{1}{20} ln 4 x^{5} - 5 x^{4} + C

Solution steps

\int \frac{x ^{4} - x ^{3}}{4 x ^{5} - 5 x ^{4}} d x

Apply u-substitution

= \int \frac{1}{20 u} d u

Take the constant out:

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

= \frac{1}{20} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{20} ln ∣ u ∣

Substitute back

u = 4 x^{5} - 5 x^{4}

= \frac{1}{20} ln 4 x^{5} - 5 x^{4}

Add a constant to the solution

= \frac{1}{20} ln 4 x^{5} - 5 x^{4} + C

What is the integral of (x^4-x^3)/(4x^5-5x^4) ?
The integral of (x^4-x^3)/(4x^5-5x^4) is 1/20 ln|4x^5-5x^4|+C