What is the integral of xe^{-125x} ?

The integral of xe^{-125x} is 1/15625 (-125e^{-125x}x-e^{-125x})+C

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integral of xe^{-125x}

\int x e^{- 125 x} d x

\frac{1}{15625} (- 125 e^{- 125 x} x - e^{- 125 x}) + C

Solution steps

\int x e^{- 125 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{15625} d u

= \int \frac{e ^{u} u}{15625} d u

Take the constant out:

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

= \frac{1}{15625} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{15625} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{15625} (e^{u} u - e^{u})

Substitute back

u = - 125 x

= \frac{1}{15625} (e^{- 125 x} (- 125 x) - e^{- 125 x})

Simplify

= \frac{1}{15625} (- 125 e^{- 125 x} x - e^{- 125 x})

Add a constant to the solution

= \frac{1}{15625} (- 125 e^{- 125 x} x - e^{- 125 x}) + C

x^{2} + x + 23, 2 x^{2} + 3

n = 5 \sum \infty \frac{1}{2 7 ^{n}}

x \to 0 lim (sin (sin (x)))

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

\int_{7}^{9} (x^{2} + 2 x - 3) d x

What is the integral of xe^{-125x} ?
The integral of xe^{-125x} is 1/15625 (-125e^{-125x}x-e^{-125x})+C