What is the integral from 0 to infinity of xe^{-3x} ?

The integral from 0 to infinity of xe^{-3x} is 1/9

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integral from 0 to infinity of xe^{-3x}

\int_{0}^{\infty} x e^{- 3 x} d x

\frac{1}{9}

Decimal Notation

0.11111 \dots

Solution steps

\int_{0}^{\infty} x e^{- 3 x} d x

Compute the indefinite integral:

\int x e^{- 3 x} d x = \frac{1}{9} (- 3 e^{- 3 x} x - e^{- 3 x}) + C

Compute the boundaries:

\int_{0}^{\infty} x e^{- 3 x} d x = 0 - (- \frac{1}{9})

= 0 - (- \frac{1}{9})

Simplify

= \frac{1}{9}

x \to 0 + lim (csc (x) - \frac{1}{x})

f (x) = - \frac{1}{x}, at x = 4

x \to 4 lim (\frac{12 - 3 x}{x - 4})

y^{^{''}} - 8 y^{^{'}} + 13 y = 0, y (0) = 0, y^{^{'}} (0) = 7

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

What is the integral from 0 to infinity of xe^{-3x} ?
The integral from 0 to infinity of xe^{-3x} is 1/9