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

The integral of sin(x)*e^{3x} is -(e^{3x}cos(x))/(10)+(3e^{3x}sin(x))/(10)+C

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

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

- \frac{e ^{3 x} cos ( x )}{10} + \frac{3 e ^{3 x} sin ( x )}{10} + C

Solution steps

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

Apply Integration By Parts

: - e^{3 x} cos (x) - \int - 3 e^{3 x} cos (x) d x

= - e^{3 x} cos (x) - \int - 3 e^{3 x} cos (x) d x

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= - \frac{e ^{3 x} cos ( x )}{10} + \frac{3 e ^{3 x} sin ( x )}{10}

Add a constant to the solution

= - \frac{e ^{3 x} cos ( x )}{10} + \frac{3 e ^{3 x} sin ( x )}{10} + C

\int_{0}^{\infty} e^{- 10 x} d x

x \to 0 + lim (∣ x ∣)

f (t) = - 3 t^{2} + 2 t - 4

\frac{d}{d x} (ln (1 + x^{2}) - arcsin (\frac{1}{x}))

\int (12 x^{11} - \frac{11}{x ^{11}}) d x

What is the integral of sin(x)*e^{3x} ?
The integral of sin(x)*e^{3x} is -(e^{3x}cos(x))/(10)+(3e^{3x}sin(x))/(10)+C