Question

The value of \(\int e^{\sin x}\sin 2x\ dx\) is

• A. 2 e^sinx (sin x - 1) + C
• B. 2 e^sinx (sin x + 1) + C
• C. 2 e^sinx (cos x + 1) + C
• D. 2 e^sinx (cos x - 1) + C

Answer 1

The Correct Option is A

\( I=Ae^{\sin(x)}\sin (x)+ Be^{\sin(x)}\cos (x)dx\) 

Then derivatve \(I= A(cos(x)e^{\sin(x)} sin x+(cosx dx e^{\sin x)}) + ... I=Ae^{\sin(x)}\sin (2x)dx \).

Consider the option A: derivative is \(2 (cos(x) e^{\sin x} sin x + e^{sin(x)} (- cos(x) dy/dx )\) \(2e^{\sin(x)) sinxcosx}\) \( + C \)

The derivative of (A) is \(2e^{\sin x} cosx( sin x-1)dx = sin (2 x dx )\) ...  

Therefore, the correct option is (A) 2 \(e^{\sin x} (sin x - 1) + C\)