What is the integral of e^x(sin(x)-cos(x)) ?

The integral of e^x(sin(x)-cos(x)) is -e^xcos(x)+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(sin(x)-cos(x))

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

- e^{x} cos (x) + C

Solution steps

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

Expand

e^{x} (sin (x) - cos (x)) : e^{x} sin (x) - e^{x} cos (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

= - e^{x} cos (x)

Add a constant to the solution

= - e^{x} cos (x) + C

\int cos (x) (6 sin^{2} (x) - 1) d x

\int \frac{x ^{2} - x - 6}{x ^{3} + x} d x

\int (x^{2} + 3 x^{13})^{5} d x

\int sin (x) - x^{4} d x

\int 1 - u^{2} d u

What is the integral of e^x(sin(x)-cos(x)) ?
The integral of e^x(sin(x)-cos(x)) is -e^xcos(x)+C