Question

Evaluate the integral: \[ \int \left[ \log(1+\cos x) - x \tan\left(\frac{x}{2}\right) \right] dx \]

• A. \( x \log|x| + C \)
• B. \( x \log|1+\sin x| + C \)
• C. \( x \log\left|\tan\frac{x}{2}\right| + C \)
• D. \( x \log|1+\cos x| + C \)

Hint:

When an integrand contains a logarithmic function and a product involving \(x\), always check if it matches the derivative of a product like \(x \log f(x)\).

Answer 1

The Correct Option is D

Step 1: Rewrite the integrand.
We are given the integral \[ \int \left[ \log(1+\cos x) - x \tan\left(\frac{x}{2}\right) \right] dx \] Split the integral into two parts: \[ \int \log(1+\cos x)\,dx - \int x \tan\left(\frac{x}{2}\right)\,dx \] Step 2: Use the identity for tangent.
Recall that \[ \tan\left(\frac{x}{2}\right) = \frac{\sin x}{1+\cos x} \] Thus, the second term becomes \[ \int x \frac{\sin x}{1+\cos x}\,dx \] Step 3: Apply differentiation insight.
Observe that \[ \frac{d}{dx}\left[\log(1+\cos x)\right] = -\frac{\sin x}{1+\cos x} \] Hence, the given integrand matches the derivative structure of \[ x \log(1+\cos x) \] Step 4: Write the final result.
Therefore, the integral evaluates to \[ x \log|1+\cos x| + C \]