What is the integral of x/(3e^{2x)} ?

The integral of x/(3e^{2x)} is 1/3 (-1/2 e^{-2x}x-1/4 e^{-2x})+C

integral of x/(3e^{2x)}

\int \frac{x}{3 e ^{2 x}} d x

\frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x}) + C

Solution steps

\int \frac{x}{3 e ^{2 x}} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{x}{e ^{2 x}} d x

Apply Integration By Parts

= \frac{1}{3} (- \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}

= \frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x})

Add a constant to the solution

= \frac{1}{3} (- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x}) + C

\int 4 x - 8 y^{3}

\frac{d y}{d x} = 2 x + y - 2

\int 2 x (x^{2} + 7)^{4} d x

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

x \to - 6 lim (2 x + 1)

What is the integral of x/(3e^{2x)} ?
The integral of x/(3e^{2x)} is 1/3 (-1/2 e^{-2x}x-1/4 e^{-2x})+C