What is the integral of (2x)/(e^{2x)} ?

The integral of (2x)/(e^{2x)} is -e^{-2x}x-1/2 e^{-2x}+C

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integral of (2x)/(e^{2x)}

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

- e^{- 2 x} x - \frac{1}{2} e^{- 2 x} + C

Solution steps

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

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} x - \int - \frac{1}{2} e^{- 2 x} d x

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

\int - \frac{1}{2} e^{- 2 x} d x = \frac{1}{4} e^{- 2 x}

= 2 (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x})

Simplify

2 (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x}) : - e^{- 2 x} x - \frac{1}{2} e^{- 2 x}

= - e^{- 2 x} x - \frac{1}{2} e^{- 2 x}

Add a constant to the solution

= - e^{- 2 x} x - \frac{1}{2} e^{- 2 x} + C

x \to 2 + lim (g (x))

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\int \frac{1}{( x ^{2} + 5 )} d x

What is the integral of (2x)/(e^{2x)} ?
The integral of (2x)/(e^{2x)} is -e^{-2x}x-1/2 e^{-2x}+C