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

The integral of e^{5x}cos(x) is (e^{5x}sin(x))/(26)+(5e^{5x}cos(x))/(26)+C

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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What is the integral of e^{5x}cos(x) ?
The integral of e^{5x}cos(x) is (e^{5x}sin(x))/(26)+(5e^{5x}cos(x))/(26)+C