Calculus

Integrals

Medium

core

Match the LIST-I with LIST-II

LIST-I (Definite integrals) LIST-II (Value)
A. $\displaystyle\int_{-6}^{6} \vert x \vert \, dx$ I. $1$
B. $\displaystyle\int_{-\pi/2}^{\pi/2} \sin^{2017} x \, dx$ II. $36$
C. $\displaystyle\int_{0}^{\pi/4} \cos x \, dx$ III. $0$
D. $\displaystyle\int_{0}^{2} \dfrac{\sqrt{x}}{\sqrt{x}+\sqrt{2-x}} \, dx$ IV. $\dfrac{1}{\sqrt{2}}$

Choose the correct answer from the options given below:

	LIST-I (Definite integrals)		LIST-II (Value)
A.	$\displaystyle\int_{-6}^{6} \vert x \vert \, dx$	I.	$1$
B.	$\displaystyle\int_{-\pi/2}^{\pi/2} \sin^{2017} x \, dx$	II.	$36$
C.	$\displaystyle\int_{0}^{\pi/4} \cos x \, dx$	III.	$0$
D.	$\displaystyle\int_{0}^{2} \dfrac{\sqrt{x}}{\sqrt{x}+\sqrt{2-x}} \, dx$	IV.	$\dfrac{1}{\sqrt{2}}$

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

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

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

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

✅ Correct Option: 1

A: $|x|$ is even, so the value is $2\int_0^6 x\,dx=36$ (II).

B: an odd power of sine over a symmetric interval integrates to $0$ (III).

C: $\sin\frac{\pi}{4}=\frac{1}{\sqrt2}$ (IV).

D: using $\int_0^a f(x)dx=\int_0^a f(a-x)dx$ , twice the integral equals $2$ , so it is $1$ (I).

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Solution