AP EAMCET · Maths · Indefinite Integration
\[
\int \frac{\cos ^4 x}{\left(\sin ^2 x+\sin ^{-3} x \cos ^5 x\right)^3} d x=
\]
- A \(\frac{1}{5}\left(1+\cot ^5 x\right)^{-2}+C\)
- B \(\frac{1}{10}\left(1+\cot ^2 x\right)^{-5}+C\)
- C \(\frac{1}{10}\left(1+\cot ^5 x\right)^{-2}+C\)
- D \(\frac{1}{5}\left(1+\cot ^5 x\right)^{-5}+C\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{10}\left(1+\cot ^5 x\right)^{-2}+C\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Let } I=\int \frac{\cos ^4 x}{\left(\sin ^2 x+\sin ^{-3} x \cos ^5 x\right)^3} d x \\ & =\int \frac{\cos ^4 x}{\left(\sin ^2 x\right)^3\left(1+\cot ^5 x\right)^3} \\ & =\int \frac{\cot ^4 x \operatorname{cosec}^2 x}{\left(1+\cot ^5 x\right)^3} d x…
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