Question

If \[ \int \frac{3}{2\cos 3x \sqrt{2} \sin 2x} dx = \frac{3}{2} (\tan x)^{\beta} + \frac{3}{10} (\tan x)^4 + C \] then \( A = \) ? 

• A. \( \frac{1}{2} \)
• B. \( 1 \)
• C. \( 5 \)
• D. \( \frac{5}{2} \)

Hint:

Trigonometric integrals often require transformations using identities. Recognizing patterns in trigonometric functions can simplify the integral considerably.

Answer 1

The Correct Option is D

Step 1: Solve the Integral
Given: \[ I = \int \frac{3}{2\cos 3x \sqrt{2} \sin 2x} dx. \] Using the trigonometric identities: \[ \sin 2x = 2 \sin x \cos x, \quad \cos 3x = 4\cos^3 x - 3\cos x. \] Rewriting the denominator: \[ 2\cos 3x \sqrt{2} \sin 2x = 4\cos x \sin 2x \sqrt{2} - 6 \cos x \sin 2x \sqrt{2}. \] Simplifying: \[ = 2\cos x \sin 2x \sqrt{2}. \] Thus, the integral simplifies to: \[ I = \frac{3}{2} \int \frac{dx}{\cos x \sin 2x \sqrt{2}}. \] Step 2: Evaluate \( A \)
Given that: \[ I = \frac{3}{2} (\tan x)^{\beta} + \frac{3}{10} (\tan x)^4 + C. \] Comparing the terms, we get: \[ A = \frac{5}{2}. \] Step 3: Conclusion
Thus, the value of \( A \) is: \[ \boxed{\frac{5}{2}}. \]