What is the integral of e^{3x}sin(e^x) ?

The integral of e^{3x}sin(e^x) is -e^{2x}cos(e^x)+2(e^xsin(e^x)+cos(e^x))+C

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integral of e^{3x}sin(e^x)

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

- e^{2 x} cos (e^{x}) + 2 (e^{x} sin (e^{x}) + cos (e^{x})) + C

Solution steps

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

Apply u-substitution

: \int u^{2} sin (u) d u

= \int u^{2} sin (u) d u

Apply Integration By Parts

: - u^{2} cos (u) - \int - 2 u cos (u) d u

= - u^{2} cos (u) - \int - 2 u cos (u) d u

\int - 2 u cos (u) d u = - 2 (u sin (u) + cos (u))

= - u^{2} cos (u) - (- 2 (u sin (u) + cos (u)))

Substitute back

u = e^{x}

= - (e^{x})^{2} cos (e^{x}) - (- 2 (e^{x} sin (e^{x}) + cos (e^{x})))

Simplify

- (e^{x})^{2} cos (e^{x}) - (- 2 (e^{x} sin (e^{x}) + cos (e^{x}))) : - e^{2 x} cos (e^{x}) + 2 (e^{x} sin (e^{x}) + cos (e^{x}))

= - e^{2 x} cos (e^{x}) + 2 (e^{x} sin (e^{x}) + cos (e^{x}))

Add a constant to the solution

= - e^{2 x} cos (e^{x}) + 2 (e^{x} sin (e^{x}) + cos (e^{x})) + C

x \to - \infty lim (2^{x})

n \to \infty lim (\frac{( - 1 ) ^{n + 1}}{( n + 3 ) \cdot n !})

\int x^{3} x^{2} - 8 d x

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

What is the integral of e^{3x}sin(e^x) ?
The integral of e^{3x}sin(e^x) is -e^{2x}cos(e^x)+2(e^xsin(e^x)+cos(e^x))+C