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

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

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

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

e^{x} + C

Solution steps

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

Simplify

x \frac{e ^{x}}{x} : \frac{e ^{x}}{x}

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

Apply u-substitution

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

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

Simplify

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

= e^{x}

Add a constant to the solution

= e^{x} + C

\frac{d}{d x} (x e^{3 z (x)})

f (x) = x^{3} + \frac{2}{x}, at x = 1

δ (t)

y^{^{'}} - 2 y = 5

\int \frac{ln ( 6 x )}{x} d x

What is the integral of sqrt(x)*(e^{sqrt(x)})/(2x) ?
The integral of sqrt(x)*(e^{sqrt(x)})/(2x) is e^{sqrt(x)}+C