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AP EAMCET · Maths · Indefinite Integration

\(\int \frac{\sec ^2 x}{\sin ^7 x} d x-\int \frac{7}{\sin ^7 x} d x=\)

  1. A \(\frac{1}{\sin ^6 x \cos x}+c\)
  2. B \(\frac{\tan x}{\sin ^8 x}+c\)
  3. C \(\sin ^8 x \cos x+c\)
  4. D \(\sec x \tan ^7 x+c\)
Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{1}{\sin ^6 x \cos x}+c\)

Step-by-step Solution

Detailed explanation

\( \int \frac{\sec ^2 x}{\sin ^7 x} d x-\int \frac{7}{\sin ^7 x} d x \) Let \( I_1 = \int \frac{\sec ^2 x}{\sin ^7 x} d x \). Using integration by parts: \( u = \frac{1}{\sin ^7 x} = \csc ^7 x \), \( dv = \sec ^2 x d x \). \( du = -7 \csc ^7 x \cot x d x \), \( v = \tan x \).…