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

The integral of e^{5x}sin(5x) is -(e^{5x}cos(5x))/(10)+(e^{5x}sin(5x))/(10)+C

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

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

- \frac{e ^{5 x} cos ( 5 x )}{10} + \frac{e ^{5 x} sin ( 5 x )}{10} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

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

= - \frac{e ^{5 x} cos ( 5 x )}{10} + \frac{e ^{5 x} sin ( 5 x )}{10}

Add a constant to the solution

= - \frac{e ^{5 x} cos ( 5 x )}{10} + \frac{e ^{5 x} sin ( 5 x )}{10} + C

y = cos (x), x = \frac{π}{4}, x = \frac{π}{3}

\frac{d y}{d x} + y = 0

\int \frac{1}{x ( 1 + x )} d x

\frac{\partial}{\partial x} (2^{x})

\int_{- 3}^{3} \frac{1}{x ^{2} + 9} d x

What is the integral of e^{5x}sin(5x) ?
The integral of e^{5x}sin(5x) is -(e^{5x}cos(5x))/(10)+(e^{5x}sin(5x))/(10)+C