Question

Evaluate the integral: \[ \int \frac{dx}{4 + 3\cot x} \]

• A. \( -\frac{3}{25} \log |4 + 3\cot x| + \frac{4}{25} x + C \)
• B. \( -\frac{3}{25} \log |4\sin x + 3\cos x| + \frac{4}{25} x + C \)
• C. \( \frac{4}{25} \log |4\sin x + 3\cos x| - \frac{3}{25} x + C \)
• D. \( \frac{4}{25} \log |4 + 3\cot x| - \frac{3}{25} x + C \)

Hint:

For integrals involving \( \cot x \), try rewriting in terms of sine and cosine, then use substitution for simplification.

Answer 1

The Correct Option is B

Step 1: Substituting Trigonometric Identities We rewrite \( 4 + 3\cot x \) using the sine and cosine functions: \[ 4 + 3\cot x = 4 + 3 \frac{\cos x}{\sin x} \] Multiplying numerator and denominator by \( \sin x \), we obtain: \[ = \frac{4\sin x + 3\cos x}{\sin x} \] Thus, the given integral becomes: \[ I = \int \frac{\sin x \, dx}{4\sin x + 3\cos x} \] 

Step 2: Using Substitution Let: \[ u = 4\sin x + 3\cos x \] Differentiating both sides: \[ du = (4\cos x - 3\sin x) dx \] We rewrite the integral using substitution: \[ I = \int \frac{dx}{4 + 3\cot x} \] Using standard integral properties: \[ I = -\frac{3}{25} \log |4\sin x + 3\cos x| + \frac{4}{25} x + C \]