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

The integral of e^x(sin(x)+cos(x)) is e^xsin(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of e^x(sin(x)+cos(x))

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

e^{x} sin (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} sin (x)

= e^{x} sin (x)

Add a constant to the solution

= e^{x} sin (x) + C

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