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

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

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

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

e^{x} cos (e^{x}) - sin (e^{x}) + C

Solution steps

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

Take the constant out:

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

= - \int e^{2 x} sin (e^{x}) d x

Apply u-substitution

: \int u sin (u) d u

= - \int u sin (u) d u

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

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

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

= - (- u cos (u) - (- sin (u)))

Substitute back

u = e^{x}

= - (- e^{x} cos (e^{x}) - (- sin (e^{x})))

Simplify

= e^{x} cos (e^{x}) - sin (e^{x})

Add a constant to the solution

= e^{x} cos (e^{x}) - sin (e^{x}) + C

ln (x + 7)

\frac{d}{d x} (\frac{x}{3} - 1)

\frac{\partial}{\partial x} (x^{3} e^{2 x y})

\int - \frac{4 cos ( t )}{3 - 10 sin ( t )} d t

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