What is the integral of (e^x*cos(x)) ?

The integral of (e^x*cos(x)) is (e^xsin(x))/2+(e^xcos(x))/2+C

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integral of (e^x*cos(x))

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

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

Solution steps

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int \frac{2}{x ln ( 8 x )} d x

\int \frac{1}{( 1 - x ) ^{\frac{3}{2}}} d x

\int e^{- \frac{2 x}{3}} d x

\int \frac{1 - x ^{2}}{x ^{2} + 1} d x

What is the integral of (e^x*cos(x)) ?
The integral of (e^x*cos(x)) is (e^xsin(x))/2+(e^xcos(x))/2+C