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

The integral of e^t(sin(t)+cos(t)) is e^tsin(t)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

e^{t} sin (t) + C

Solution steps

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

Expand

e^{t} (sin (t) + cos (t)) : e^{t} sin (t) + e^{t} cos (t)

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

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

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

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

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

Simplify

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

= e^{t} sin (t)

Add a constant to the solution

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

\int_{- 3}^{2} x^{2} d x

\int \frac{1}{4 sin ( x ) - 3 cos ( x )} d x

\int_{1}^{\infty} \frac{1}{x ^{0}} d x

(1 - \frac{2}{y} + x) \frac{d y}{d x} + y = 2 x - 1

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

What is the integral of e^t(sin(t)+cos(t)) ?
The integral of e^t(sin(t)+cos(t)) is e^tsin(t)+C