JEE Main202229 Jul 2022Morning ShiftMathematicsDefinite IntegrationActual
The integral ∫ 0 π 2 1 3 + 2 sin x + cos x d x is equal to:
Options
- Atan - 1 2
- Btan - 1 2 - π 4
- C1 2 tan - 1 2 - π 8
- D1 2
Correct answer
B. tan - 1 2 - π 4
Step-by-step solution
Given, I = ∫ 0 π 2 d x 3 + 2 sin x + cos x ⇒ I = ∫ 0 π 2 d x 3 + 2 2 tan x 2 1 + tan 2 x 2 + 1 - tan 2 x 2 1 + tan 2 x 2 ⇒ I = ∫ 0 π 2 sec 2 x 2 · d x 2 tan 2 x 2 + 4 tan x 2 + 4 Put tan x 2 = t   ⇒ 1 2 sec 2 x 2 d x = d t , so ⇒ I = ∫ 0 1 d t t + 1 2 + 1 ⇒ I = tan - 1 t + 1 0 1 ⇒ I = tan - 1 2 - π 4