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

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

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

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

2 (- \frac{sin ( x )}{e ^{s i n (x)}} - \frac{1}{e ^{s i n (x)}}) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u}{e ^{u}} d u

= 2 \cdot \int \frac{u}{e ^{u}} d u

Apply Integration By Parts

: - \frac{u}{e ^{u}} - \int - \frac{1}{e ^{u}} d u

= 2 (- \frac{u}{e ^{u}} - \int - \frac{1}{e ^{u}} d u)

\int - \frac{1}{e ^{u}} d u = \frac{1}{e ^{u}}

= 2 (- \frac{u}{e ^{u}} - \frac{1}{e ^{u}})

Substitute back

u = sin (x)

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

Add a constant to the solution

= 2 (- \frac{sin ( x )}{e ^{s i n (x)}} - \frac{1}{e ^{s i n (x)}}) + C

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