What is the integral of e^xcos(x) ?

The integral of e^xcos(x) is (e^xsin(x))/2+(e^xcos(x))/2+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^xcos(x)

integral

e^{x} cos (x)

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

Solution steps

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int \frac{x ^{4}}{( 9 + x ^{5} ) ^{2}} d x

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

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

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

\int 36 (x^{2} + 1) (x^{3} + 3 x)^{2} d x

What is the integral of e^xcos(x) ?
The integral of e^xcos(x) is (e^xsin(x))/2+(e^xcos(x))/2+C