Match List-I with List-II

Choose the correct answer from the options given below.

(A)-(II), (B)-(III), (C)-(IV), (D)-(I)

The correct answer is Option (1) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

(A) ∫ from 1 to e of $\frac{\log x}{x}\,dx$

Put $u=\log x,\; du=\frac{1}{x}dx$

$=\int_{0}^{1} u\,du = \frac{1}{2}$

(B) ∫ from −2 to 2 of $x^{3}(1-x^{2})\,dx$

$x^{3}$ is odd and $(1-x^{2})$ is even ⇒ product is odd ⇒ integral over symmetric limits is $0$

(C) ∫ from 1 to 2 of $x\,dx$

$=\frac{x^{2}}{2}\Big|_{1}^{2}=\frac{4-1}{2}=\frac{3}{2}$

(D) ∫ from −2 to 2 of $|x|\,dx$

$=2\int_{0}^{2} x\,dx=2\cdot \frac{2^{2}}{2}=4$

Thus, the correct matching is: A–II, B–III, C–IV, D–I.