What is the integral of e^xcos(wx) ?

The integral of e^xcos(wx) is (we^xsin(wx))/(w^2+1)+(e^xcos(wx))/(w^2+1)+C

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integral of e^xcos(wx)

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

\frac{w e ^{x} sin ( w x )}{w ^{2} + 1} + \frac{e ^{x} cos ( w x )}{w ^{2} + 1} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{w e ^{x} sin ( w x )}{w ^{2} + 1} + \frac{e ^{x} cos ( w x )}{w ^{2} + 1}

Add a constant to the solution

= \frac{w e ^{x} sin ( w x )}{w ^{2} + 1} + \frac{e ^{x} cos ( w x )}{w ^{2} + 1} + C

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What is the integral of e^xcos(wx) ?
The integral of e^xcos(wx) is (we^xsin(wx))/(w^2+1)+(e^xcos(wx))/(w^2+1)+C