What is the integral of e^{5x}sin(6x) ?

The integral of e^{5x}sin(6x) is -(6e^{5x}cos(6x))/(61)+(5e^{5x}sin(6x))/(61)+C

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integral of e^{5x}sin(6x)

\int e^{5 x} sin (6 x) d x

- \frac{6 e ^{5 x} cos ( 6 x )}{61} + \frac{5 e ^{5 x} sin ( 6 x )}{61} + C

Solution steps

\int e^{5 x} sin (6 x) d x

Apply Integration By Parts

: - \frac{1}{6} e^{5 x} cos (6 x) - \int - \frac{5}{6} e^{5 x} cos (6 x) d x

= - \frac{1}{6} e^{5 x} cos (6 x) - \int - \frac{5}{6} e^{5 x} cos (6 x) d x

Take the constant out:

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

= - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} \cdot \int e^{5 x} cos (6 x) d x)

Apply Integration By Parts

: \frac{1}{6} e^{5 x} sin (6 x) - \int \frac{5}{6} e^{5 x} sin (6 x) d x

= - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} (\frac{1}{6} e^{5 x} sin (6 x) - \int \frac{5}{6} e^{5 x} sin (6 x) d x))

Take the constant out:

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

= - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} (\frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} \cdot \int e^{5 x} sin (6 x) d x))

Therefore

\int e^{5 x} sin (6 x) d x = - \frac{1}{6} e^{5 x} cos (6 x) - (- \frac{5}{6} (\frac{1}{6} e^{5 x} sin (6 x) - \frac{5}{6} \cdot \int e^{5 x} sin (6 x) d x))

Isolate

\int e^{5 x} sin (6 x) d x

= - \frac{6 e ^{5 x} cos ( 6 x )}{61} + \frac{5 e ^{5 x} sin ( 6 x )}{61}

Add a constant to the solution

= - \frac{6 e ^{5 x} cos ( 6 x )}{61} + \frac{5 e ^{5 x} sin ( 6 x )}{61} + C

\frac{s - 1}{s + 1}

\frac{\partial}{\partial y} (x^{2} - x y + 4 y^{2})

x \to 2 lim (2 x - 7)

\frac{d}{d x} (\frac{sin ( 2 x )}{cos ( x )})

\frac{d}{d x} (4^{c o s (x)})

What is the integral of e^{5x}sin(6x) ?
The integral of e^{5x}sin(6x) is -(6e^{5x}cos(6x))/(61)+(5e^{5x}sin(6x))/(61)+C