What is the integral of x*e^{-5x} ?

The integral of x*e^{-5x} is 1/25 (-5e^{-5x}x-e^{-5x})+C

integral of x*e^{-5x}

\int x \cdot e^{- 5 x} d x

\frac{1}{25} (- 5 e^{- 5 x} x - e^{- 5 x}) + C

Solution steps

\int x e^{- 5 x} d x

Apply u-substitution

= \int \frac{e ^{u} u}{25} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int e^{u} u d u

Apply Integration By Parts

= \frac{1}{25} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{25} (e^{u} u - e^{u})

Substitute back

u = - 5 x

= \frac{1}{25} (e^{- 5 x} (- 5 x) - e^{- 5 x})

Simplify

= \frac{1}{25} (- 5 e^{- 5 x} x - e^{- 5 x})

Add a constant to the solution

= \frac{1}{25} (- 5 e^{- 5 x} x - e^{- 5 x}) + C

\frac{d}{d x} (5^{5 x})

\int \frac{1}{3} tan (3 x) d x

\int cos^{2} (θ) - sin^{2} (θ) d θ

x \to 0 lim ((1 - 8 x)^{\frac{1}{x}})

\frac{d y}{d x} = e^{5 y} sin (6 x)

What is the integral of x*e^{-5x} ?
The integral of x*e^{-5x} is 1/25 (-5e^{-5x}x-e^{-5x})+C