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AP EAMCET · Maths · Indefinite Integration

\(\int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x=\)

  1. A \(\log (\sin 5 x \sin 2 x)+c\)
  2. B \(\log \sin 5 x+\sin 2 x+c\)
  3. C \(\frac{1}{5} \log \sin 5 x+\frac{1}{2} \log \sin 2 x+c\)
  4. D \(\frac{1}{5} \log \sin x+\frac{1}{2} \log \sin x+c\)
Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{1}{5} \log \sin 5 x+\frac{1}{2} \log \sin 2 x+c\)

Step-by-step Solution

Detailed explanation

\begin{aligned} & I=\int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x \Rightarrow I=\int \frac{\sin (5 x+2 x)}{\sin 2 x \sin 5 x} d x \\ & \Rightarrow I=\int \frac{\sin 5 x \cos 2 x+\cos 5 x \sin 2 x}{\sin 2 x \sin 5 x} d x \\ & \Rightarrow I=\int \cot 2 x d x+\int \cot 5 x d x \\ &…