What is the integral of e^{x/2}cos(3x) ?

The integral of e^{x/2}cos(3x) is 2(6/37 e^{x/2}sin(3x)+1/37 e^{x/2}cos(3x))+C

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integral of e^{x/2}cos(3x)

\int e^{\frac{x}{2}} cos (3 x) d x

2 (\frac{6}{37} e^{\frac{x}{2}} sin (3 x) + \frac{1}{37} e^{\frac{x}{2}} cos (3 x)) + C

Solution steps

\int e^{\frac{x}{2}} cos (3 x) d x

Apply u-substitution

: \int \frac{1}{3} e^{\frac{u}{6}} cos (u) d u

= \int \frac{1}{3} e^{\frac{u}{6}} cos (u) d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{\frac{u}{6}} cos (u) d u

Apply u-substitution

: \int e^{v} cos (6 v) \cdot 6 d v

= \frac{1}{3} \cdot \int e^{v} cos (6 v) \cdot 6 d v

Take the constant out:

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

= \frac{1}{3} \cdot 6 \cdot \int e^{v} cos (6 v) d v

Apply Integration By Parts

: e^{v} \frac{1}{6} sin (6 v) - \int e^{v} \frac{1}{6} sin (6 v) d v

= \frac{1}{3} \cdot 6 (e^{v} \frac{1}{6} sin (6 v) - \int e^{v} \frac{1}{6} sin (6 v) d v)

Take the constant out:

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

= \frac{1}{3} \cdot 6 (e^{v} \frac{1}{6} sin (6 v) - \frac{1}{6} \cdot \int e^{v} sin (6 v) d v)

Apply Integration By Parts

: - \frac{1}{6} e^{v} cos (6 v) - \int - \frac{1}{6} e^{v} cos (6 v) d v

= \frac{1}{3} \cdot 6 (e^{v} \frac{1}{6} sin (6 v) - \frac{1}{6} (- \frac{1}{6} e^{v} cos (6 v) - \int - \frac{1}{6} e^{v} cos (6 v) d v))

Take the constant out:

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

= \frac{1}{3} \cdot 6 (e^{v} \frac{1}{6} sin (6 v) - \frac{1}{6} (- \frac{1}{6} e^{v} cos (6 v) - (- \frac{1}{6} \cdot \int e^{v} cos (6 v) d v)))

Therefore

6 \cdot \frac{1}{3} \cdot \int e^{v} cos (6 v) d v = \frac{1}{3} \cdot 6 (e^{v} \frac{1}{6} sin (6 v) - \frac{1}{6} (- \frac{1}{6} e^{v} cos (6 v) - (- \frac{1}{6} \cdot \int e^{v} cos (6 v) d v)))

Isolate

\int e^{v} cos (6 v) d v

= 6 \cdot \frac{1}{3} (\frac{6 e ^{v} sin ( 6 v )}{37} + \frac{e ^{v} cos ( 6 v )}{37})

Substitute back

= 6 \cdot \frac{1}{3} (\frac{6 e ^{\frac{3 x}{6}} sin ( 6 \cdot \frac{3 x}{6} )}{37} + \frac{e ^{\frac{3 x}{6}} cos ( 6 \cdot \frac{3 x}{6} )}{37})

Simplify

6 \cdot \frac{1}{3} (\frac{6 e ^{\frac{3 x}{6}} sin ( 6 \cdot \frac{3 x}{6} )}{37} + \frac{e ^{\frac{3 x}{6}} cos ( 6 \cdot \frac{3 x}{6} )}{37}) : 2 (\frac{6}{37} e^{\frac{x}{2}} sin (3 x) + \frac{1}{37} e^{\frac{x}{2}} cos (3 x))

= 2 (\frac{6}{37} e^{\frac{x}{2}} sin (3 x) + \frac{1}{37} e^{\frac{x}{2}} cos (3 x))

Add a constant to the solution

= 2 (\frac{6}{37} e^{\frac{x}{2}} sin (3 x) + \frac{1}{37} e^{\frac{x}{2}} cos (3 x)) + C

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What is the integral of e^{x/2}cos(3x) ?
The integral of e^{x/2}cos(3x) is 2(6/37 e^{x/2}sin(3x)+1/37 e^{x/2}cos(3x))+C