Answer to Question #167811 in Calculus for Phyroe

Question

Evaluate the integral of y² Csc y³ dy

Answer 1

$\int y^2csc(y^3) dy =[u=y^3,du=3y^2dy]=\frac{1}{3} \int csc(u)du = \frac{1}{3} \int \frac{csc(u)(cot(u)+csc(u))}{cot(u)+csc(u)}du=\frac{1}{3} \int -\frac{-csc(u)cot(u)-csc^2(u)}{cot(u)+csc(u)}du=[s=cot(u)+csc(u), ds=(-csc^2(u)-csc(u)cot(u))du]=-\frac{1}{3} \int \frac{1}{s}ds=-\frac{ln|s|}{3}+C=-\frac{ln|cot(u)+csc(u)|}{3}+C=-\frac{ln|cot(y^3)+csc(y^3)|}{3}+C=[cot(y^3)+csc(y^3)=\frac{1}{sinx}+\frac{cosx}{sinx}=\frac{1+cosx}{sinx}=cot(\frac{x}{2})]=-\frac{ln|cot(\frac{y^3}{2})|}{3}+C=-\frac{ln|\frac{cos(\frac{y^3}{2})}{sin(\frac{y^3}{2})}|}{3}+C=-\frac{ln|cos(\frac{y^3}{2})|-ln|sin(\frac{y^3}{2})|}{3}+C=\frac{ln|sin(\frac{y^3}{2})|-ln|cos(\frac{y^3}{2})|}{3}+C$