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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int cos (u) e^{s i n (u)} \frac{1}{2} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int e^{v} d v

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

Use the common integral:

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

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

Substitute back

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

Simplify

2 \cdot \frac{1}{2} e^{s i n (2 x)} : 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 2cos(2x)e^{sin(2x)} ?
The integral of 2cos(2x)e^{sin(2x)} is e^{sin(2x)}+C