What is the integral of 1/2 sin(2t)e^{sin(t)} ?

The integral of 1/2 sin(2t)e^{sin(t)} is e^{sin(t)}sin(t)-e^{sin(t)}+C

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integral of 1/2 sin(2t)e^{sin(t)}

\int \frac{1}{2} sin (2 t) e^{s i n (t)} d t

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

Solution steps

\int \frac{1}{2} sin (2 t) e^{s i n (t)} d t

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (2 t) e^{s i n (t)} d t

Rewrite using trig identities

= \frac{1}{2} \cdot \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

= \frac{1}{2} \cdot 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

= \frac{1}{2} \cdot 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)}

= \frac{1}{2} \cdot 2 (sin (t) e^{s i n (t)} - e^{s i n (t)})

Simplify

\frac{1}{2} \cdot 2 (sin (t) e^{s i n (t)} - e^{s i n (t)}) : e^{s i n (t)} sin (t) - e^{s i n (t)}

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

Add a constant to the solution

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

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What is the integral of 1/2 sin(2t)e^{sin(t)} ?
The integral of 1/2 sin(2t)e^{sin(t)} is e^{sin(t)}sin(t)-e^{sin(t)}+C