What is the integral of 1/2 xe^{x/2} ?

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

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integral of 1/2 xe^{x/2}

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = \frac{x}{2}

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

Simplify

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

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

Add a constant to the solution

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

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