Question

The value of \(\int (\log \sec x) \tan x \,dx\) is

• A. \(\sec x + c\)
• B. \(\log \sec x + c\)
• C. \(\frac{1}{2}(\log \sec x)^2 + c\)
• D. \(\log(\log \sec x) + c\)

Hint:

In substitution integrals, look for a composite function. The derivative of the "inner" function is often present as a factor in the integrand. Here, the inner function is \(\log(\sec x)\), and its derivative, \(\tan x\), is conveniently the other factor. Recognizing this pattern is key.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Find ∫sin²x dx.

Step 2: Key Formula (Alternate):
Use power-reduction: sin²x = (1-cos 2x)/2.

Step 3: Detailed Explanation:
∫(1-cos 2x)/2 dx = (1/2)∫dx - (1/2)∫cos 2x dx = x/2 - (sin 2x)/4 + c.

Step 4: Final Answer:
Integral is x/2 - (sin 2x)/4 + c.