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

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

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

integral

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

x^{3} \cdot (e^{- x})^{2}

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

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

\frac{1}{( x ( x ^{2} + x + 1 ) )}

5 (1 - x)^{4}

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