What is the integral of 3e^xcos(5x) ?

The integral of 3e^xcos(5x) is 3((5e^xsin(5x))/(26)+(e^xcos(5x))/(26))+C

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integral of 3e^xcos(5x)

\int 3 e^{x} cos (5 x) d x

3 (\frac{5 e ^{x} sin ( 5 x )}{26} + \frac{e ^{x} cos ( 5 x )}{26}) + C

Solution steps

\int 3 e^{x} cos (5 x) d x

Take the constant out:

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

= 3 \cdot \int e^{x} cos (5 x) d x

Apply Integration By Parts

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

= 3 (e^{x} \frac{1}{5} sin (5 x) - \int e^{x} \frac{1}{5} sin (5 x) d x)

Take the constant out:

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

= 3 (e^{x} \frac{1}{5} sin (5 x) - \frac{1}{5} \cdot \int e^{x} sin (5 x) d x)

Apply Integration By Parts

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

= 3 (e^{x} \frac{1}{5} sin (5 x) - \frac{1}{5} (- \frac{1}{5} e^{x} cos (5 x) - \int - \frac{1}{5} e^{x} cos (5 x) d x))

Take the constant out:

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

= 3 (e^{x} \frac{1}{5} sin (5 x) - \frac{1}{5} (- \frac{1}{5} e^{x} cos (5 x) - (- \frac{1}{5} \cdot \int e^{x} cos (5 x) d x)))

Therefore

3 \cdot \int e^{x} cos (5 x) d x = 3 (e^{x} \frac{1}{5} sin (5 x) - \frac{1}{5} (- \frac{1}{5} e^{x} cos (5 x) - (- \frac{1}{5} \cdot \int e^{x} cos (5 x) d x)))

Isolate

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

= 3 (\frac{5 e ^{x} sin ( 5 x )}{26} + \frac{e ^{x} cos ( 5 x )}{26})

Add a constant to the solution

= 3 (\frac{5 e ^{x} sin ( 5 x )}{26} + \frac{e ^{x} cos ( 5 x )}{26}) + C

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What is the integral of 3e^xcos(5x) ?
The integral of 3e^xcos(5x) is 3((5e^xsin(5x))/(26)+(e^xcos(5x))/(26))+C