What is the integral of-sin(e^x)e^{2x} ?

The integral of-sin(e^x)e^{2x} is e^xcos(e^x)-sin(e^x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of-sin(e^x)e^{2x}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= - \int u sin (u) d u

Apply Integration By Parts

= - (- u cos (u) - \int - cos (u) d u)

\int - cos (u) d u = - sin (u)

= - (- u cos (u) - (- sin (u)))

Substitute back

u = e^{x}

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

Simplify

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

Add a constant to the solution

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

What is the integral of-sin(e^x)e^{2x} ?
The integral of-sin(e^x)e^{2x} is e^xcos(e^x)-sin(e^x)+C