Question

$\int\frac{cot x}{\sqrt[3]{sin x}} dx =$

• A. $\frac{-3}{\sqrt[3]{sin x} } + C$
• B. $\frac{-2 }{sin^{3} x}+ C$
• C. $\frac{3}{sin^{\frac{1}{3}} x}+ C$
• D. None of these

Answer 1

Answer: (A)

$I = \int\frac{cot\, x}{\sqrt[3]{sin x}} dx =\int\frac{cos\, x}{sin^{\frac{1}{3}} x \cdot sin\, x} dx $

$= \int\frac{cos x }{sin^{\frac{4}{3}} x} dx = \int sin ^{-\frac{4}{3}} x \cdot cos x $dx

Put $sin x=t \Rightarrow cos x dx = dt $

$\Rightarrow I= \int t^{-\frac{4}{3}}dt =\frac{ t^{-\frac{1}{3}}}{-\frac{1}{3}} + C =\frac{-3}{\sqrt[3]{sin x}} + C$