What is the integral of (2x^2)/(5x^3+70) ?

The integral of (2x^2)/(5x^3+70) is 2/15 ln|5x^3+70|+C

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integral of (2x^2)/(5x^3+70)

\int \frac{2 x ^{2}}{5 x ^{3} + 70} d x

\frac{2}{15} ln 5 x^{3} + 70 + C

Solution steps

\int \frac{2 x ^{2}}{5 x ^{3} + 70} 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}}{5 x ^{3} + 70} d x

Apply u-substitution

: \int \frac{1}{15 u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{15} ln ∣ u ∣

Substitute back

u = 5 x^{3} + 70

= 2 \cdot \frac{1}{15} ln 5 x^{3} + 70

Simplify

2 \cdot \frac{1}{15} ln 5 x^{3} + 70 : \frac{2}{15} ln 5 x^{3} + 70

= \frac{2}{15} ln 5 x^{3} + 70

Add a constant to the solution

= \frac{2}{15} ln 5 x^{3} + 70 + C

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What is the integral of (2x^2)/(5x^3+70) ?
The integral of (2x^2)/(5x^3+70) is 2/15 ln|5x^3+70|+C