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AP EAMCET · Maths · Indefinite Integration

If \(I_n=\int \frac{\sin n x}{\cos x} d x\), then \(I_n=\)

  1. A \(\frac{-2}{n-1} \cos (n-1) x-\mathrm{I}_{n-2}\)
  2. B \(\frac{2}{n-1} \cos (n-1) x+\mathrm{I}_{n-2}\)
  3. C \(\frac{-2}{n+1} \sin (n+1) x-\mathrm{I}_{n-2}\)
  4. D \(\frac{-2}{n+1} \cos (n-1) x-\mathrm{I}_{n-2}\)
Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{-2}{n-1} \cos (n-1) x-\mathrm{I}_{n-2}\)

Step-by-step Solution

Detailed explanation

\(\begin{aligned}= & \int \frac{\sin (n x-x+x)}{\cos x} d x \\ = & \int \frac{\sin [(n-1) x+x]}{\cos x} d x \\ & \int \frac{\sin (n-1) x \cos x+\cos (n-1) x \sin x}{\cos } d x\end{aligned}\)…