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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u}}{4} d u

= 2 \cdot \int \frac{e ^{u}}{4} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int e^{u} d u

Use the common integral:

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

= 2 \cdot \frac{1}{4} e^{u}

Substitute back

u = 2 sin (2 x)

= 2 \cdot \frac{1}{4} e^{2 s i n (2 x)}

Simplify

2 \cdot \frac{1}{4} e^{2 s i n (2 x)} : \frac{1}{2} e^{2 s i n (2 x)}

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

Add a constant to the solution

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

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