Question

evaluate the integral using any appropriate algebraic method or trigonometric identity.
int\frac{dy}{27sqrt3{y^{2}}(1 + y^{2/3})}
int\frac{dy}{27sqrt3{y^{2}}(1 + y^{2/3})}=square

Explanation:

Step1: Use substitution

Let $u = 1 + y^{2/3}$, then $du=\frac{2}{3}y^{- 1/3}dy$. The integral $\int\frac{dy}{27\sqrt[3]{y^{2}}(1 + y^{2/3})}=\frac{1}{18}\int\frac{du}{u}$.

Step2: Integrate

$\frac{1}{18}\int\frac{du}{u}=\frac{1}{18}\ln|u|+C$.

Step3: Substitute back

Substitute $u = 1 + y^{2/3}$ back, we get $\frac{1}{18}\ln|1 + y^{2/3}|+C$.

Answer:

$\frac{1}{18}\ln|1 + y^{2/3}|+C$