What is the integral of x*a*e^{-ax} ?

The integral of x*a*e^{-ax} is -e^{-ax}x-(e^{-ax})/a+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of xae^{-ax}

\int x \cdot a \cdot e^{- a x} d x

- e^{- a x} x - \frac{e ^{- a x}}{a} + C

Solution steps

\int x a e^{- a x} d x

Take the constant out:

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

= a \cdot \int x e^{- a x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{a ^{2}} d u

= a \cdot \int \frac{e ^{u} u}{a ^{2}} d u

Take the constant out:

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

= a \frac{1}{a ^{2}} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= a \frac{1}{a ^{2}} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= a \frac{1}{a ^{2}} (e^{u} u - e^{u})

Substitute back

u = - a x

= a \frac{1}{a ^{2}} (e^{- a x} (- a x) - e^{- a x})

Simplify

a \frac{1}{a ^{2}} (e^{- a x} (- a x) - e^{- a x}) : - e^{- a x} x - \frac{e ^{- a x}}{a}

= - e^{- a x} x - \frac{e ^{- a x}}{a}

Add a constant to the solution

= - e^{- a x} x - \frac{e ^{- a x}}{a} + C

\int cos (x) e^{5 + 2 s i n (x)} d x

\int e^{8 x} 8 d x

\int (1 + 6 x)^{4} d x

\int sec^{3} (θ) tan^{3} (θ) d θ

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

What is the integral of x*a*e^{-ax} ?
The integral of x*a*e^{-ax} is -e^{-ax}x-(e^{-ax})/a+C