What is the integral of e^3cos(2x)sin(2x) ?

The integral of e^3cos(2x)sin(2x) is -(e^3)/8 cos(4x)+C

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integral of e^3cos(2x)sin(2x)

\int e^{3} cos (2 x) sin (2 x) d x

- \frac{e ^{3}}{8} cos (4 x) + C

Solution steps

\int e^{3} cos (2 x) sin (2 x) d x

Take the constant out:

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

= e^{3} \cdot \int cos (2 x) sin (2 x) d x

Rewrite using trig identities

= e^{3} \cdot \int \frac{1}{2} sin (4 x) d x

Take the constant out:

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

= e^{3} \frac{1}{2} \cdot \int sin (4 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{4} d u

= e^{3} \frac{1}{2} \cdot \int sin (u) \frac{1}{4} d u

Take the constant out:

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

= e^{3} \frac{1}{2} \cdot \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= e^{3} \frac{1}{2} \cdot \frac{1}{4} (- cos (u))

Substitute back

u = 4 x

= e^{3} \frac{1}{2} \cdot \frac{1}{4} (- cos (4 x))

Simplify

e^{3} \frac{1}{2} \cdot \frac{1}{4} (- cos (4 x)) : - \frac{e ^{3}}{8} cos (4 x)

= - \frac{e ^{3}}{8} cos (4 x)

Add a constant to the solution

= - \frac{e ^{3}}{8} cos (4 x) + C

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What is the integral of e^3cos(2x)sin(2x) ?
The integral of e^3cos(2x)sin(2x) is -(e^3)/8 cos(4x)+C