What is the integral of (v^2)/(v^3+100) ?

The integral of (v^2)/(v^3+100) is 1/3 ln|v^3+100|+C

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integral of (v^2)/(v^3+100)

\int \frac{v ^{2}}{v ^{3} + 100} d v

\frac{1}{3} ln v^{3} + 100 + C

Solution steps

\int \frac{v ^{2}}{v ^{3} + 100} d v

Apply u-substitution

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = v^{3} + 100

= \frac{1}{3} ln v^{3} + 100

Add a constant to the solution

= \frac{1}{3} ln v^{3} + 100 + C

What is the integral of (v^2)/(v^3+100) ?
The integral of (v^2)/(v^3+100) is 1/3 ln|v^3+100|+C