Question

Evaluate the integral: ∫12log⁡xdx

• A. (A) 2log⁡2−1
• B. (B) 2log⁡2+1
• C. (C) 2log⁡2−3
• D. (D) None of these

Answer 1

Answer: (A)

(A) 2log⁡2−1

Answer 2

Explanation:
Let's first integrate the expression under integral by parts.I=∫log⁡xdx=∫(1)(log⁡x)dxConsidering log⁡x as the first function and 1 as the second function, we get:=(log⁡x)∫1dx−∫[1x∫1dx]dx [∵∫f(x)g(x)dx=f(x)∫g(x)dx−∫[f′(x)∫g(x)dx]dx=(log⁡x)x−x+CPutting the limits of the definite integral, we get:∫12log⁡xdx=[x(log⁡x)−x]12=(2log⁡2−2)−(0−1)=2log⁡2−1Hence, the correct option is (A).