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} \cdot 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} (\frac{5 x yz}{x ^{2} + y ^{2} + z ^{2}})

y^{^{'}} + 2 y = 10 sin (t)

f (x) = 3 x^{2} + 5

\frac{\partial}{\partial t} (cos (3 t))

\frac{x ^{2}}{2 ( 1 - x )}

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