What is the integral of e^{-x}2x ?

The integral of e^{-x}2x is 2(-e^{-x}x-e^{-x})+C

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integral of e^{-x}2x

\int e^{- x} 2 x d x

2 (- e^{- x} x - e^{- x}) + C

Solution steps

\int e^{- x} \cdot 2 x d x

Take the constant out:

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

= 2 \cdot \int e^{- x} x d x

Apply Integration By Parts

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

= 2 (- e^{- x} x - \int - e^{- x} d x)

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

= 2 (- e^{- x} x - e^{- x})

Add a constant to the solution

= 2 (- e^{- x} x - e^{- x}) + C

(4 x y + y^{2}) d x + (2 x^{2} + 2 x y) d y = 0

\frac{d y}{d x} = \frac{x - 1}{y - 1}

\int (x + 5)^{3} d x

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

x \to 0 lim (\frac{- x ^{2}}{x ^{2}})

What is the integral of e^{-x}2x ?
The integral of e^{-x}2x is 2(-e^{-x}x-e^{-x})+C