Definite Integration
Periodic function integrals
Grade Class 12
Question:
Let F(x) be an indefinite integral of sin^2 x.<br>Statement-1: The function F(x) satisfies F(x + π) = F(x) for all real x.<br>because<br>Statement-2: sin^2(x + π) = sin^2 x for all real x.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(C) Statement-1 is True, Statement-2 is False.
(D) Statement-1 is False, Statement-2 is True.
Step-by-Step Solution
Key Concept: Integration of periodic functions
F(x) = \int sin^2 x dx = \int (1 - cos 2x)/2 dx = x/2 - sin 2x / 4 + C. F(x + \pi) = (x + \pi)/2 - sin(2x + 2\pi)/4 + C = x/2 + \pi/2 - sin 2x / 4 + C = F(x) + \pi/2. So Statement-1 is False.
Correct Answer: D