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

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

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

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

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

Solution steps

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

Rewrite using trig identities

= \int sin (2 x) e^{\frac{1 - c o s ( 2 x )}{2}} d x

Apply u-substitution

: \int \frac{e ^{\frac{u}{2}}}{2} d u

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

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{\frac{u}{2}} d u

Apply u-substitution

: \int e^{v} \cdot 2 d v

= \frac{1}{2} \cdot \int e^{v} \cdot 2 d v

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^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= \frac{1}{2} \cdot 2 e^{v}

Substitute back

= \frac{1}{2} \cdot 2 e^{\frac{1 - c o s ( 2 x )}{2}}

Simplify

\frac{1}{2} \cdot 2 e^{\frac{1 - c o s ( 2 x )}{2}} : e^{s i n^{2} (x)}

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

Add a constant to the solution

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

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