What is the integral of e^x*cos(3x) ?

The integral of e^x*cos(3x) is (3e^xsin(3x))/(10)+(e^xcos(3x))/(10)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^x*cos(3x)

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

\frac{3 e ^{x} sin ( 3 x )}{10} + \frac{e ^{x} cos ( 3 x )}{10} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{3 e ^{x} sin ( 3 x )}{10} + \frac{e ^{x} cos ( 3 x )}{10}

Add a constant to the solution

= \frac{3 e ^{x} sin ( 3 x )}{10} + \frac{e ^{x} cos ( 3 x )}{10} + C

\int \frac{6 x ^{2} - 15 x + 22}{( x + 3 ) ( x ^{2} + 2 ) ^{2}} d x

y = \frac{1}{x}, (4, \frac{1}{2})

y^{^{''}} - 3 y^{^{'}} + 2 y = e^{3 t}

\frac{\partial}{\partial x} (x^{s i n (x)})

6 \cdot (\frac{d y}{d x}) = y^{2}, y (0) = 4

What is the integral of e^x*cos(3x) ?
The integral of e^x*cos(3x) is (3e^xsin(3x))/(10)+(e^xcos(3x))/(10)+C