Question

The area bounded by the curves y = f (x), the X-axis and the ordinates x = 1 and x = b is (b - 1 ) sin (3b + 4). Then, f (x) is equal to

• A. (a) (x - 1) cos (3x + 4)
• B. (b) 8sin (3x + 4)
• C. (c) sin (3x + 4) + 3(x - 1) cos (3x + 4)
• D. (d) None of the above

Answer 1

Answer: (C)

(c) sin (3x + 4) + 3(x - 1) cos (3x + 4)

Answer 2

Since, $\int \limits_1^b f(x) dx = (b - 1) sin (3b + 4)$
On differentiating both sides w.r.t. b, we get
$f(b) = 3(b - 1). cos(3b + 4) + sin(3b + 4)$
$\therefore f(x) = sin(3x + 4) + 3(x - 1) cos(3x + 4)$