What is the integral of e^{5x}cos(x/3) ?

The integral of e^{5x}cos(x/3) is 3(1/226 e^{5x}sin(x/3)+15/226 e^{5x}cos(x/3))+C

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integral of e^{5x}cos(x/3)

\int e^{5 x} cos (\frac{x}{3}) d x

3 (\frac{1}{226} e^{5 x} sin (\frac{x}{3}) + \frac{15}{226} e^{5 x} cos (\frac{x}{3})) + C

Solution steps

\int e^{5 x} cos (\frac{x}{3}) d x

Apply u-substitution

: \int 3 e^{15 u} cos (u) d u

= \int 3 e^{15 u} cos (u) d u

Take the constant out:

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

= 3 \cdot \int e^{15 u} cos (u) d u

Apply Integration By Parts

: e^{15 u} sin (u) - \int e^{15 u} \cdot 15 sin (u) d u

= 3 (e^{15 u} sin (u) - \int e^{15 u} \cdot 15 sin (u) d u)

Take the constant out:

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

= 3 (e^{15 u} sin (u) - 15 \cdot \int e^{15 u} sin (u) d u)

Apply Integration By Parts

: - e^{15 u} cos (u) - \int - 15 e^{15 u} cos (u) d u

= 3 (e^{15 u} sin (u) - 15 (- e^{15 u} cos (u) - \int - 15 e^{15 u} cos (u) d u))

Take the constant out:

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

= 3 (e^{15 u} sin (u) - 15 (- e^{15 u} cos (u) - (- 15 \cdot \int e^{15 u} cos (u) d u)))

Therefore

3 \cdot \int e^{15 u} cos (u) d u = 3 (e^{15 u} sin (u) - 15 (- e^{15 u} cos (u) - (- 15 \cdot \int e^{15 u} cos (u) d u)))

Isolate

\int e^{15 u} cos (u) d u

= 3 (\frac{e ^{15 u} sin ( u )}{226} + \frac{15 e ^{15 u} cos ( u )}{226})

Substitute back

u = \frac{x}{3}

= 3 (\frac{e ^{15 \cdot \frac{x}{3}} sin ( \frac{x}{3} )}{226} + \frac{15 e ^{15 \cdot \frac{x}{3}} cos ( \frac{x}{3} )}{226})

Simplify

3 (\frac{e ^{15 \cdot \frac{x}{3}} sin ( \frac{x}{3} )}{226} + \frac{15 e ^{15 \cdot \frac{x}{3}} cos ( \frac{x}{3} )}{226}) : 3 (\frac{1}{226} e^{5 x} sin (\frac{x}{3}) + \frac{15}{226} e^{5 x} cos (\frac{x}{3}))

= 3 (\frac{1}{226} e^{5 x} sin (\frac{x}{3}) + \frac{15}{226} e^{5 x} cos (\frac{x}{3}))

Add a constant to the solution

= 3 (\frac{1}{226} e^{5 x} sin (\frac{x}{3}) + \frac{15}{226} e^{5 x} cos (\frac{x}{3})) + C

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What is the integral of e^{5x}cos(x/3) ?
The integral of e^{5x}cos(x/3) is 3(1/226 e^{5x}sin(x/3)+15/226 e^{5x}cos(x/3))+C