Question

\(\int \frac{1+\cos a\cos x}{\cos a+\cos x}dx=\)

• A. \(\sin a\log |\frac{\cot \frac{a}{2}+\tan \frac{x}{2}}{\cot \frac{a}{2}-\tan \frac{x}{2}}|+x\cos a+c\)
• B. \(\sin a\log |\frac{\cot \frac{a}{2}-\tan \frac{x}{2}}{\cot \frac{a}{2}+\tan \frac{x}{2}}|+x\cos a+c\)
• C. \(\sin a\log |\frac{\cot \frac{a}{2}+\tan \frac{x}{2}}{\cot \frac{a}{2}-\tan \frac{x}{2}}|+x\sin a+c\)
• D. \(\sin a\log |\frac{\cot \frac{a}{2}-\tan \frac{x}{2}}{\cot \frac{a}{2}+\tan \frac{x}{2}}|+x\sin a+c\)

Hint:

Rewrite the numerator using the denominator, then use the substitution t equal to tan of x over 2.

Answer 1

The Correct Option is A

Step 1: Check the polynomial part
The term outside the logarithm comes from the constant $\cos a$ in the split, so it must be $x\cos a$. This rules out options C and D.

Step 2: Check the logarithm part
With $\cot\frac{a}{2}=k$, $\int\frac{2\,dt}{(1+\cos a)(k^{2}-t^{2})}$ gives $\frac{1}{2k}\log\left|\frac{k+t}{k-t}\right|$ times a positive constant, so the plus sign is in the numerator.

Step 3: Conclude
This rules out option B and leaves option A.

Final Answer:
The correct result is the first option. \[ \boxed{\text{(A) }\sin a\log \left|\dfrac{\cot \frac{a}{2}+\tan \frac{x}{2}}{\cot \frac{a}{2}-\tan \frac{x}{2}}\right|+x\cos a+c} \]