What is the integral of e^{sin(x)}sin(2x) ?

The integral of e^{sin(x)}sin(2x) is 2(sin(x)e^{sin(x)}-e^{sin(x)})+C

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integral of e^{sin(x)}sin(2x)

\int e^{s i n (x)} sin (2 x) d x

2 (sin (x) e^{s i n (x)} - e^{s i n (x)}) + C

Solution steps

\int e^{s i n (x)} sin (2 x) d x

Rewrite using trig identities

= \int e^{s i n (x)} \cdot 2 cos (x) sin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int e^{s i n (x)} cos (x) sin (x) d x

Apply Integration By Parts

= 2 (sin (x) e^{s i n (x)} - \int cos (x) e^{s i n (x)} d x)

\int cos (x) e^{s i n (x)} d x = e^{s i n (x)}

= 2 (sin (x) e^{s i n (x)} - e^{s i n (x)})

Add a constant to the solution

= 2 (sin (x) e^{s i n (x)} - e^{s i n (x)}) + C

What is the integral of e^{sin(x)}sin(2x) ?
The integral of e^{sin(x)}sin(2x) is 2(sin(x)e^{sin(x)}-e^{sin(x)})+C