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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int e^{u} \cdot 2 d u

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{x}{2}

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

Simplify

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

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

Add a constant to the solution

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

n = 0 \sum \infty (- \frac{5}{8})^{n}

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