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

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

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integral of e^{1+sin(2x)}cos(2x)

\int e^{1 + s i n (2 x)} cos (2 x) d x

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

Solution steps

\int e^{1 + s i n (2 x)} cos (2 x) d x

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = 1 + sin (2 x)

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

Add a constant to the solution

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

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