Question

If \(I = \int \frac{\sin x + \sin^3 x}{\cos 2x} dx = P \cos x + Q \log \left| \frac{\sqrt{2} \cos x - 1}{\sqrt{2} \cos x + 1} \right|\) (where \(c\) is a constant of integration), then values of \(P\) and \(Q\) are respectively

• A. \(\frac{1}{2}\), \(\frac{3}{4\sqrt{2}}\)
• B. \(\frac{1}{2}\), \(\frac{-3}{4\sqrt{2}}\)
• C. \(\frac{1}{2}\), \(\frac{3}{2\sqrt{2}}\)
• D. \(\frac{1}{2}\), \(\frac{-3}{2\sqrt{2}}\)

Hint:

Whenever the integrand contains \(\sin x\) and \(\cos x\) where one has an odd power, try substituting the other function. Here, \(\sin x\) is essentially factored out, making \(\cos x = t\) the ideal substitution.

Answer 1

The Correct Option is B