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

The integral of e^{3x}cos(x/3) is 3(1/82 e^{3x}sin(x/3)+9/82 e^{3x}cos(x/3))+C

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

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

3 (\frac{1}{82} e^{3 x} sin (\frac{x}{3}) + \frac{9}{82} e^{3 x} cos (\frac{x}{3})) + C

Solution steps

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

Apply u-substitution

: \int 3 e^{9 u} cos (u) d u

= \int 3 e^{9 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^{9 u} cos (u) d u

Apply Integration By Parts

: e^{9 u} sin (u) - \int e^{9 u} \cdot 9 sin (u) d u

= 3 (e^{9 u} sin (u) - \int e^{9 u} \cdot 9 sin (u) d u)

Take the constant out:

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

= 3 (e^{9 u} sin (u) - 9 \cdot \int e^{9 u} sin (u) d u)

Apply Integration By Parts

: - e^{9 u} cos (u) - \int - 9 e^{9 u} cos (u) d u

= 3 (e^{9 u} sin (u) - 9 (- e^{9 u} cos (u) - \int - 9 e^{9 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^{9 u} sin (u) - 9 (- e^{9 u} cos (u) - (- 9 \cdot \int e^{9 u} cos (u) d u)))

Therefore

3 \cdot \int e^{9 u} cos (u) d u = 3 (e^{9 u} sin (u) - 9 (- e^{9 u} cos (u) - (- 9 \cdot \int e^{9 u} cos (u) d u)))

Isolate

\int e^{9 u} cos (u) d u

= 3 (\frac{e ^{9 u} sin ( u )}{82} + \frac{9 e ^{9 u} cos ( u )}{82})

Substitute back

u = \frac{x}{3}

= 3 (\frac{e ^{9 \cdot \frac{x}{3}} sin ( \frac{x}{3} )}{82} + \frac{9 e ^{9 \cdot \frac{x}{3}} cos ( \frac{x}{3} )}{82})

Simplify

3 (\frac{e ^{9 \cdot \frac{x}{3}} sin ( \frac{x}{3} )}{82} + \frac{9 e ^{9 \cdot \frac{x}{3}} cos ( \frac{x}{3} )}{82}) : 3 (\frac{1}{82} e^{3 x} sin (\frac{x}{3}) + \frac{9}{82} e^{3 x} cos (\frac{x}{3}))

= 3 (\frac{1}{82} e^{3 x} sin (\frac{x}{3}) + \frac{9}{82} e^{3 x} cos (\frac{x}{3}))

Add a constant to the solution

= 3 (\frac{1}{82} e^{3 x} sin (\frac{x}{3}) + \frac{9}{82} e^{3 x} cos (\frac{x}{3})) + C

4 tan (θ)

\frac{d}{d x} (\frac{( sin ( \frac{1}{x} ) \cdot x ^{3} )}{ln ( x )})

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

\int_{0}^{\frac{11}{12}} 24 x d x

\frac{\partial}{\partial x} (x^{\frac{2}{3}} \cdot y^{\frac{1}{2}})

What is the integral of e^{3x}cos(x/3) ?
The integral of e^{3x}cos(x/3) is 3(1/82 e^{3x}sin(x/3)+9/82 e^{3x}cos(x/3))+C