What is the integral of xe^{-2} ?

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

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integral of xe^{-2}

\int x e^{- 2} d x

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

Solution steps

\int x e^{- 2} d x

Take the constant out:

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

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

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

Apply the Power Rule

: \frac{x ^{2}}{2}

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

Simplify

\frac{1}{e ^{2}} \cdot \frac{x ^{2}}{2} : \frac{x ^{2}}{2 e ^{2}}

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

Add a constant to the solution

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

f (x) = 4 + 8 x - 5 x^{2}

e^{e^{x^{2}}}

(x^{3} - y^{3}) d x + (x y^{2}) d y = 0

\frac{d y}{d x} = 3 (y + 7) x^{2}

\frac{1 - 3 t}{6 + t}