What is the integral of 1/(sqrt(x)e^{\sqrt{x)}} ?

The integral of 1/(sqrt(x)e^{\sqrt{x)}} is (2sqrt(x))/e+C

integral of 1/(sqrt(x)e^{\sqrt{x)}}

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

\frac{2 x}{e} + C

Solution steps

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

Apply exponent rule:

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

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

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

Take the constant out:

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

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

Apply the Power Rule

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

Simplify

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

= \frac{2 x}{e}

Add a constant to the solution

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

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What is the integral of 1/(sqrt(x)e^{\sqrt{x)}} ?
The integral of 1/(sqrt(x)e^{\sqrt{x)}} is (2sqrt(x))/e+C