99 Percentile Qs Bank for JEE MainMathematicsDefinite Integration
If f x = A sin π x 2 + B ,   f ' 1 2 = 2   and ∫ 0 1 f x d x = 2 A π , then constants A and B are respectively
Options
- A0   and   − 4 π
- B4 π   and  0
- C2 π   and   3 π
- Dπ 2   and   π 2
Correct answer
B. 4 π   and  0
Step-by-step solution
f ( x ) = A sin ( π x 2 ) + B ⇒ f ⁡ ' x = A π 2 cos   π x 2 ⇒   f ⁡ ′ 1 2 = A π 2 cos π 4 = 2 ⇒ A = 4 π and ∫ 0 1 f ( x ) d x = 2 A π ⇒ ∫ 0 1 [ A sin ( π x 2 ) + B ] d x = 2 A π ⇒ [ − 2 A π cos ( π x 2 ) + B x ] 0 1 = 2 A π ⇒ B + 2 A π = 2 A π ⇒ B = 0