Let ∫( sin 3 θ+ sin θ) cos θ ⋅ e sin θ ⋅ d θ=(A ⋅ sin 3 θ+B ⋅ cos 2 θ+C ⋅ sin θ+D ⋅ cos θ+E) ⋅ e sin θ+K where K is constant of integration. Find the value of (B/A)+(D/E).

Question

Let ∫( sin 3 θ+ sin θ) cos θ ⋅ e sin θ ⋅ d θ=(A ⋅ sin 3 θ+B ⋅ cos 2 θ+C ⋅ sin θ+D ⋅ cos θ+E) ⋅ e sin θ+K where K is constant of integration. Find the value of (B/A)+(D/E). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let I=∫ 2 ⋅ sin 2 θ ⋅ cos 2 θ ⋅ e sin θ d θ=∫ 4 sin θ ⋅ cos 3 θ ⋅ e sin θ ⋅ d θ Put sin θ=t, so I =∫ 4 t ( l - t 2) e t dt =4 ∫ I t ⋅ e t -4 ∫ t 3 ⋅ e t dt ∴ I =4 I 1-4 I 2 text We get, I =-4 sin 3 θ-12 cos 2 θ-20 sin θ+32 ∴ A =-4, B =-12, C =-20, D =0, E =32

Q. Let ∫(sin3θ+sinθ)cosθ⋅esinθ⋅dθ=(A⋅sin3θ+B⋅cos2θ+C⋅sinθ+D⋅cosθ+E)⋅esinθ+K where K is constant of integration. Find the value of AB​+ED​.

Let I=∫2⋅sin2θ⋅cos2θ⋅esinθdθ=∫4sinθ⋅cos3θ⋅esinθ⋅dθ Put sinθ=t, so I=∫4t(l−t2)etdt=4∫It​⋅et−4∫t3⋅etdt ∴I=4I1​−4I2​ We get, I=−4sin3θ−12cos2θ−20sinθ+32 ∴A=−4,B=−12,C=−20,D=0,E=32

Q. Let $\int (sin 3 θ + sin θ) cos θ \cdot e^{s i n θ} \cdot d θ = (A \cdot sin^{3} θ + B \cdot cos^{2} θ + C \cdot sin θ + D \cdot cos θ + E) \cdot e^{s i n θ} + K$
where $K$ is constant of integration. Find the value of $\frac{B}{A} + \frac{D}{E}$ .