If \(f(x) = A \sin(x/2) + B\) , \(f'(\pi/2) = 2\) , and \(\sqrt{2}\int_0^1 f(x)dx = \frac{2A}{\pi}\) , then the constants A and B are

Indefinite Integration

Finding Constants

Grade 12

Question:

If \(f(x) = A \sin(x/2) + B\), \(f'(\pi/2) = 2\), and \(\sqrt{2}\int_0^1 f(x)dx = \frac{2A}{\pi}\), then the constants A and B are

(A) \(\pi/2\) and \(\pi/2\)

(B) \(2\pi\) and \(3\pi\)

(C) 0 and \(-4/\pi\)

(D) \(8/\pi\) and 0

Key Concept: Use the derivative condition and the integral condition to set up a system of equations for A and B.

IIT - 1995

Correct Answer: D