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

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

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integral of sin(2t)*e^{sin(t)}

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

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

Solution steps

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

Rewrite using trig identities

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

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 (t)} cos (t) sin (t) d t

Apply Integration By Parts

: sin (t) e^{s i n (t)} - \int cos (t) e^{s i n (t)} d t

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

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

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

Add a constant to the solution

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

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