What is the integral of (2x^2)/(20-5x^3) ?

The integral of (2x^2)/(20-5x^3) is -2/15 ln|20-5x^3|+C

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integral of (2x^2)/(20-5x^3)

\int \frac{2 x ^{2}}{20 - 5 x ^{3}} d x

- \frac{2}{15} ln 20 - 5 x^{3} + C

Solution steps

\int \frac{2 x ^{2}}{20 - 5 x ^{3}} d x

Take the constant out:

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

= 2 \cdot \int \frac{x ^{2}}{20 - 5 x ^{3}} d x

Apply u-substitution

= 2 \cdot \int - \frac{1}{15 u} d u

Take the constant out:

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

= 2 (- \frac{1}{15} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{15} ln ∣ u ∣)

Substitute back

u = 20 - 5 x^{3}

= 2 (- \frac{1}{15} ln 20 - 5 x^{3})

Simplify

2 (- \frac{1}{15} ln 20 - 5 x^{3}) : - \frac{2}{15} ln 20 - 5 x^{3}

= - \frac{2}{15} ln 20 - 5 x^{3}

Add a constant to the solution

= - \frac{2}{15} ln 20 - 5 x^{3} + C

What is the integral of (2x^2)/(20-5x^3) ?
The integral of (2x^2)/(20-5x^3) is -2/15 ln|20-5x^3|+C