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

The integral of ((e^x)(xe^x))/(-2) is -1/8 (2e^{2x}x-e^{2x})+C

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

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

- \frac{1}{8} (2 e^{2 x} x - e^{2 x}) + C

Solution steps

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

Take the constant out:

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

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

Simplify

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

Simplify

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

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

Apply u-substitution

: \int \frac{e ^{u} u}{4} d u

= (- \frac{1}{2}) \cdot \int \frac{e ^{u} u}{4} d u

Take the constant out:

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

= (- \frac{1}{2}) \frac{1}{4} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= (- \frac{1}{2}) \frac{1}{4} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

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

Substitute back

u = 2 x

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

Simplify

(- \frac{1}{2}) \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) : - \frac{1}{8} (2 e^{2 x} x - e^{2 x})

= - \frac{1}{8} (2 e^{2 x} x - e^{2 x})

Add a constant to the solution

= - \frac{1}{8} (2 e^{2 x} x - e^{2 x}) + C

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What is the integral of ((e^x)(xe^x))/(-2) ?
The integral of ((e^x)(xe^x))/(-2) is -1/8 (2e^{2x}x-e^{2x})+C