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

The integral of (e^{2x}cos(3x)) is (3e^{2x}sin(3x))/(13)+(2e^{2x}cos(3x))/(13)+C

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

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

\frac{3 e ^{2 x} sin ( 3 x )}{13} + \frac{2 e ^{2 x} cos ( 3 x )}{13} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

= \frac{1}{3} e^{2 x} sin (3 x) - \frac{2}{3} \cdot \int e^{2 x} sin (3 x) d x

Apply Integration By Parts

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

= \frac{1}{3} e^{2 x} sin (3 x) - \frac{2}{3} (- \frac{1}{3} e^{2 x} cos (3 x) - \int - \frac{2}{3} e^{2 x} cos (3 x) d x)

Take the constant out:

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

= \frac{1}{3} e^{2 x} sin (3 x) - \frac{2}{3} (- \frac{1}{3} e^{2 x} cos (3 x) - (- \frac{2}{3} \cdot \int e^{2 x} cos (3 x) d x))

Therefore

\int e^{2 x} cos (3 x) d x = \frac{1}{3} e^{2 x} sin (3 x) - \frac{2}{3} (- \frac{1}{3} e^{2 x} cos (3 x) - (- \frac{2}{3} \cdot \int e^{2 x} cos (3 x) d x))

Isolate

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

= \frac{3 e ^{2 x} sin ( 3 x )}{13} + \frac{2 e ^{2 x} cos ( 3 x )}{13}

Add a constant to the solution

= \frac{3 e ^{2 x} sin ( 3 x )}{13} + \frac{2 e ^{2 x} cos ( 3 x )}{13} + C

3 x^{2} - 6 x + 5

\int \frac{cos ( x )}{1 + 2 sin ( x )} d x

\frac{d}{d x} (e^{l n (x^{2})})

x \to 0 lim (\frac{1}{x ^{3}})

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

What is the integral of (e^{2x}cos(3x)) ?
The integral of (e^{2x}cos(3x)) is (3e^{2x}sin(3x))/(13)+(2e^{2x}cos(3x))/(13)+C