What is the integral of xe^{-6x} ?

The integral of xe^{-6x} is 1/36 (-6e^{-6x}x-e^{-6x})+C

integral of xe^{-6x}

\int x e^{- 6 x} d x

\frac{1}{36} (- 6 e^{- 6 x} x - e^{- 6 x}) + C

Solution steps

\int x e^{- 6 x} d x

Apply u-substitution

= \int \frac{e ^{u} u}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int e^{u} u d u

Apply Integration By Parts

= \frac{1}{36} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{36} (e^{u} u - e^{u})

Substitute back

u = - 6 x

= \frac{1}{36} (e^{- 6 x} (- 6 x) - e^{- 6 x})

Simplify

= \frac{1}{36} (- 6 e^{- 6 x} x - e^{- 6 x})

Add a constant to the solution

= \frac{1}{36} (- 6 e^{- 6 x} x - e^{- 6 x}) + C

\frac{d}{d x} (2 x^{2} cos (6 x))

\int csc (2 x) d x

\frac{\partial}{\partial x} (x^{8} y)

\frac{d}{d x} (\frac{3 x - 1}{x ^{2} + 3})

\frac{d}{d x} (\frac{2}{sin ^{2} ( 2 x )})

What is the integral of xe^{-6x} ?
The integral of xe^{-6x} is 1/36 (-6e^{-6x}x-e^{-6x})+C