Let ∫ x 3 sin x d x = g ( x ) + C , where C is the constant of integration. If 8 ( g ( 2 π ) + g ′ ( 2 π ) ) = α π 3 + β π 2 + γ , α , β , γ ∈ Z , then α + β − γ equals :

Question

Let ∫ x 3 sin x d x = g ( x ) + C , where C is the constant of integration. If 8 ( g ( 2 π ​ ) + g ′ ( 2 π ​ ) ) = α π 3 + β π 2 + γ , α , β , γ ∈ Z , then α + β − γ equals :

TestHub · Accepted Answer

​ ∫ x 3 sin x d x = − x 3 cos x + ∫ 3 x 2 cos x d x = − x 3 cos x + 3 x 2 sin x − ∫ 6 x sin x d x = − x 3 cos x + 3 x 2 sin x + 6 x cos x − 6 sin x + c ​ So g ( x ) = − x 3 cos x + 3 x 2 sin x + 6 x cos x − 6 sin x ​ g ( 2 π ​ ) = 4 3 π 2 ​ − 6 g ′ ( x ) = − 3 x 2 cos x + x 3 sin x + 6 cos x − 6 cos x g ′ ( 2 π ​ ) = 8 π 3 ​ 8 ( g ( 2 π ​ ) + g ′ ( 2 π ​ ) ) = π 3 + 6 π 2 − 48 ​ So α + β − γ = 55

Mathematics - Indefinite Integration Question with Solution | TestHub

Options:

Doubts & Discussion