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

The integral of e^{5x}cos(3x) is (3e^{5x}sin(3x))/(34)+(5e^{5x}cos(3x))/(34)+C

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

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

\frac{3 e ^{5 x} sin ( 3 x )}{34} + \frac{5 e ^{5 x} cos ( 3 x )}{34} + C

Solution steps

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

Apply Integration By Parts

= \frac{1}{3} e^{5 x} sin (3 x) - \int \frac{5}{3} e^{5 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^{5 x} sin (3 x) - \frac{5}{3} \cdot \int e^{5 x} sin (3 x) d x

Apply Integration By Parts

= \frac{1}{3} e^{5 x} sin (3 x) - \frac{5}{3} (- \frac{1}{3} e^{5 x} cos (3 x) - \int - \frac{5}{3} e^{5 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^{5 x} sin (3 x) - \frac{5}{3} (- \frac{1}{3} e^{5 x} cos (3 x) - (- \frac{5}{3} \cdot \int e^{5 x} cos (3 x) d x))

Therefore

\int e^{5 x} cos (3 x) d x = \frac{1}{3} e^{5 x} sin (3 x) - \frac{5}{3} (- \frac{1}{3} e^{5 x} cos (3 x) - (- \frac{5}{3} \cdot \int e^{5 x} cos (3 x) d x))

Isolate

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

= \frac{3 e ^{5 x} sin ( 3 x )}{34} + \frac{5 e ^{5 x} cos ( 3 x )}{34}

Add a constant to the solution

= \frac{3 e ^{5 x} sin ( 3 x )}{34} + \frac{5 e ^{5 x} cos ( 3 x )}{34} + C

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

l a pl a ce t r an s f or m t \cdot sin (k \cdot t)

x \to - 4 lim (- 5 x - 8)

x y^{^{'}} - 2 y = x^{2} - x - 2

y^{^{'}} = x e^{y - x^{2}}

What is the integral of e^{5x}cos(3x) ?
The integral of e^{5x}cos(3x) is (3e^{5x}sin(3x))/(34)+(5e^{5x}cos(3x))/(34)+C