Question

The value of the integral \(\int \frac{1}{\cos x\cdot \log \left(\tan \left(\frac{\pi }{4}+\frac{x}{2}\right)\right)}dx\) is....

• A. \(\csc \left(\frac{x}{2}\right)+\cot \left(\frac{x}{2}\right)+c\)
• B. \(\sec ^{2}(2x)+\log x+c\)
• C. \(\log |\log |\tan \left(\frac{\pi }{4}+\frac{x}{2}\right)||+c\)
• D. \(\tan \left[\log |\sec \left(\frac{\pi }{4}+\frac{x}{2}\right)|\right]+c\)

Hint:

Substitute t equal to the logarithm; its derivative is sec x.

Answer 1

The Correct Option is C

Step 1: Rewrite
$\dfrac1{\cos x\,\log\tan(\ldots)}=\dfrac{\sec x}{\log\tan(\ldots)}$.

Step 2: Recognise the derivative
We know $\dfrac{d}{dx}\log\tan\left(\dfrac\pi4+\dfrac x2\right)=\sec x$.

Step 3: Use the standard form
The integrand has the form $\dfrac{g'(x)}{g(x)}$, so the integral is $\log|g(x)|+c$.

Final Answer:
The integral is $\log|\log|\tan(\frac\pi4+\frac x2)||+c$. \[ \boxed{\text{(C) }\log |\log |\tan \left(\dfrac{\pi }{4}+\dfrac{x}{2}\right)||+c} \]