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

Concept: To solve this integral, we express the numerator and denominator in terms of \(\cos x\) to facilitate a substitution \(u = \cos x\).

Step 1: Simplifying the integrand.
The integral is: \[ I = \int \frac{\sin x (1 + \sin^2 x)}{2 \cos^2 x - 1} dx \] Substitute \(\sin^2 x = 1 - \cos^2 x\): \[ I = \int \frac{\sin x (2 - \cos^2 x)}{2 \cos^2 x - 1} dx \]

Step 2: Using substitution.
Let \(\cos x = t\), then \(-\sin x dx = dt \implies \sin x dx = -dt\). The integral becomes: \[ I = \int \frac{-(2 - t^2)}{2t^2 - 1} dt = \int \frac{t^2 - 2}{2t^2 - 1} dt \]

Step 3: Performing partial fraction decomposition.
Perform polynomial division or manipulation: \[ \frac{t^2 - 2}{2t^2 - 1} = \frac{1}{2} \left( \frac{2t^2 - 4}{2t^2 - 1} \right) = \frac{1}{2} \left( 1 - \frac{3}{2t^2 - 1} \right) \] Integrate: \[ I = \frac{1}{2} t - \frac{3}{2} \int \frac{1}{2t^2 - 1} dt \] Using the formula \(\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left| \frac{x-a}{x+a} \right|\): \[ I = \frac{1}{2} \cos x - \frac{3}{4} \int \frac{1}{t^2 - (1/\sqrt{2})^2} dt \] \[ I = \frac{1}{2} \cos x - \frac{3}{4} \cdot \frac{1}{2(1/\sqrt{2})} \log \left| \frac{t - 1/\sqrt{2}}{t + 1/\sqrt{2}} \right| \] \[ I = \frac{1}{2} \cos x - \frac{3}{4\sqrt{2}} \log \left| \frac{\sqrt{2} \cos x - 1}{\sqrt{2} \cos x + 1} \right| \]

Step 4: Comparing with the given form.
Comparing with \(P \cos x + Q \log | \dots |\): \[ P = \frac{1}{2}, \quad Q = -\frac{3}{4\sqrt{2}} \]