What is the integral of e^{5x}cos(9x) ?

The integral of e^{5x}cos(9x) is (9e^{5x}sin(9x))/(106)+(5e^{5x}cos(9x))/(106)+C

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integral of e^{5x}cos(9x)

\int e^{5 x} cos (9 x) d x

\frac{9 e ^{5 x} sin ( 9 x )}{106} + \frac{5 e ^{5 x} cos ( 9 x )}{106} + C

Solution steps

\int e^{5 x} cos (9 x) d x

Apply Integration By Parts

: \frac{1}{9} e^{5 x} sin (9 x) - \int \frac{5}{9} e^{5 x} sin (9 x) d x

= \frac{1}{9} e^{5 x} sin (9 x) - \int \frac{5}{9} e^{5 x} sin (9 x) d x

Take the constant out:

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

= \frac{1}{9} e^{5 x} sin (9 x) - \frac{5}{9} \cdot \int e^{5 x} sin (9 x) d x

Apply Integration By Parts

: - \frac{1}{9} e^{5 x} cos (9 x) - \int - \frac{5}{9} e^{5 x} cos (9 x) d x

= \frac{1}{9} e^{5 x} sin (9 x) - \frac{5}{9} (- \frac{1}{9} e^{5 x} cos (9 x) - \int - \frac{5}{9} e^{5 x} cos (9 x) d x)

Take the constant out:

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

= \frac{1}{9} e^{5 x} sin (9 x) - \frac{5}{9} (- \frac{1}{9} e^{5 x} cos (9 x) - (- \frac{5}{9} \cdot \int e^{5 x} cos (9 x) d x))

Therefore

\int e^{5 x} cos (9 x) d x = \frac{1}{9} e^{5 x} sin (9 x) - \frac{5}{9} (- \frac{1}{9} e^{5 x} cos (9 x) - (- \frac{5}{9} \cdot \int e^{5 x} cos (9 x) d x))

Isolate

\int e^{5 x} cos (9 x) d x

= \frac{9 e ^{5 x} sin ( 9 x )}{106} + \frac{5 e ^{5 x} cos ( 9 x )}{106}

Add a constant to the solution

= \frac{9 e ^{5 x} sin ( 9 x )}{106} + \frac{5 e ^{5 x} cos ( 9 x )}{106} + C

f (x) = \frac{2}{x}, at x = 9

\frac{\partial}{\partial z} (- \frac{1}{( y + 2 z ) ^{2}})

\int \frac{3 e ^{2 x}}{1 + e ^{2 x}} d x

f (x) = sin (2 x), 0, \frac{π}{2}

\frac{d}{d x} (5 sin (x) + 3 cos (x))

What is the integral of e^{5x}cos(9x) ?
The integral of e^{5x}cos(9x) is (9e^{5x}sin(9x))/(106)+(5e^{5x}cos(9x))/(106)+C