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

The integral of e^{2x}*cos(x) is (e^{2x}sin(x))/5+(2e^{2x}cos(x))/5+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{e ^{2 x} sin ( x )}{5} + \frac{2 e ^{2 x} cos ( x )}{5} + C

Solution steps

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

Apply Integration By Parts

: e^{2 x} sin (x) - \int e^{2 x} \cdot 2 sin (x) d x

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{e ^{2 x} sin ( x )}{5} + \frac{2 e ^{2 x} cos ( x )}{5}

Add a constant to the solution

= \frac{e ^{2 x} sin ( x )}{5} + \frac{2 e ^{2 x} cos ( x )}{5} + C

y^{^{''}} + y = x, y (0) = 2, y^{^{'}} (0) = 0

\int 3 x^{2} - 8 x d x

f (x) = 3 1 + x

\int 0.6 x + 2 d x

\int 3 x 1 - 2 x^{2} d x

What is the integral of e^{2x}*cos(x) ?
The integral of e^{2x}*cos(x) is (e^{2x}sin(x))/5+(2e^{2x}cos(x))/5+C