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

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

\frac{\partial}{\partial x} (- cos (x y))

\int \frac{1}{v} d v

\frac{2}{3 x ^{2}}

y^{^{'}} = \frac{y + x}{y - x}

y = \frac{3}{x}, y = \frac{3}{x ^{2}}, x = 7

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