What is the integral of (sqrt(y))/(sqrt(y^3)+4) ?

The integral of (sqrt(y))/(sqrt(y^3)+4) is 2/3 ln|y^{3/2}+4|+C

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integral of (sqrt(y))/(sqrt(y^3)+4)

\int \frac{y}{y ^{3} + 4} d y

\frac{2}{3} ln y^{\frac{3}{2}} + 4 + C

Solution steps

\int \frac{y}{y ^{3} + 4} d y

Apply u-substitution

: \int \frac{2}{3 u} d u

= \int \frac{2}{3 u} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{2}{3} ln ∣ u ∣

Substitute back

u = y^{3} + 4

= \frac{2}{3} ln y^{3} + 4

Apply radical rule:

assuming

a \geq 0

= \frac{2}{3} ln y^{\frac{3}{2}} + 4

Add a constant to the solution

= \frac{2}{3} ln y^{\frac{3}{2}} + 4 + C

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What is the integral of (sqrt(y))/(sqrt(y^3)+4) ?
The integral of (sqrt(y))/(sqrt(y^3)+4) is 2/3 ln|y^{3/2}+4|+C