AP EAMCET201921 Apr 2019Evening ShiftMathematicsIndefinite IntegrationActual
∫ x 2 2 sin π 4 + x + e x d x =
Options
- Ax 2 + 2 x - 2 sin x + - x 2 + 2 x + 2 cos x + x 2 - 2 x + 2 e x + C
- B- x 2 + 2 x - 2 sin x + x 2 + 2 x - 2 cos x + x 2 - 2 x + 2 e x + C
- Cx 2 + 2 x + 2 sin x + - x 2 - 2 x - 2 cos x + x 2 - 2 x + 2 e x + C
- Dx 2 - 2 x - 2 sin x + - x 2 + 2 x - 2 cos x + x 2 - 2 x + 2 e x + C
Correct answer
A. x 2 + 2 x - 2 sin x + - x 2 + 2 x + 2 cos x + x 2 - 2 x + 2 e x + C
Step-by-step solution
The integral expression is given as, I = ∫ x 2 2 sin π 4 + x + e x d x ⇒ I = ∫ x 2 2 sin π 4 cos x + sin x cos π 4 + e x d x ⇒ I = ∫ x 2 2 1 2 cos x + 1 2 sin x + e x d x Further simplifying we get, ⇒ I = ∫ x 2 cos x + sin x + e x d x ⇒ I = ∫ x 2 ( cos x + sin x ) d x + ∫ x 2 e x d x As we know from integration by parts, ∫ u v d x = u ∫ v d x - ∫ d u d x · ∫ v d x d x ⇒ I = x 2 ∫ cos x + sin x d x - &