What is the integral of e^xcos(0.5x) ?

The integral of e^xcos(0.5x) is 0.4e^xsin(0.5x)+0.8e^xcos(0.5x)+C

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integral of e^xcos(0.5x)

\int e^{x} cos (0.5 x) d x

0.4 e^{x} sin (0.5 x) + 0.8 e^{x} cos (0.5 x) + C

Solution steps

\int e^{x} cos (0.5 x) d x

Apply Integration By Parts

: e^{x} \cdot 2 sin (0.5 x) - \int e^{x} \cdot 2 sin (0.5 x) d x

= e^{x} \cdot 2 sin (0.5 x) - \int e^{x} \cdot 2 sin (0.5 x) d x

Take the constant out:

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

= e^{x} \cdot 2 sin (0.5 x) - 2 \cdot \int e^{x} sin (0.5 x) d x

Apply Integration By Parts

: - 2 e^{x} cos (0.5 x) - \int - 2 e^{x} cos (0.5 x) d x

= e^{x} \cdot 2 sin (0.5 x) - 2 (- 2 e^{x} cos (0.5 x) - \int - 2 e^{x} cos (0.5 x) d x)

Take the constant out:

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

= e^{x} \cdot 2 sin (0.5 x) - 2 (- 2 e^{x} cos (0.5 x) - (- 2 \cdot \int e^{x} cos (0.5 x) d x))

Therefore

\int e^{x} cos (0.5 x) d x = e^{x} \cdot 2 sin (0.5 x) - 2 (- 2 e^{x} cos (0.5 x) - (- 2 \cdot \int e^{x} cos (0.5 x) d x))

Isolate

\int e^{x} cos (0.5 x) d x

= \frac{2 e ^{x} sin ( 0.5 x )}{5} + \frac{4 e ^{x} cos ( 0.5 x )}{5}

Simplify

= 0.4 e^{x} sin (0.5 x) + 0.8 e^{x} cos (0.5 x)

Add a constant to the solution

= 0.4 e^{x} sin (0.5 x) + 0.8 e^{x} cos (0.5 x) + C

\frac{sin ( x )}{x}

\int \frac{x ^{p}}{p + 1} d x

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

\int \frac{( ln ( x ) ) ^{5}}{5} d x

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

What is the integral of e^xcos(0.5x) ?
The integral of e^xcos(0.5x) is 0.4e^xsin(0.5x)+0.8e^xcos(0.5x)+C