What is the integral of (e^{sqrt(x)}-5)/(sqrt(x)) ?

The integral of (e^{sqrt(x)}-5)/(sqrt(x)) is 2(e^{sqrt(x)}-5sqrt(x))+C

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integral of (e^{sqrt(x)}-5)/(sqrt(x))

\int \frac{e ^{x} - 5}{x} d x

2 (e^{x} - 5 x) + C

Solution steps

\int \frac{e ^{x} - 5}{x} d x

Apply u-substitution

: \int 2 (e^{u} - 5) d u

= \int 2 (e^{u} - 5) d u

Take the constant out:

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

= 2 \cdot \int e^{u} - 5 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int e^{u} d u - \int 5 d u)

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

\int 5 d u = 5 u

= 2 (e^{u} - 5 u)

Substitute back

u = x

= 2 (e^{x} - 5 x)

Add a constant to the solution

= 2 (e^{x} - 5 x) + C

\int \frac{sin ( 5 x )}{x} d x

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What is the integral of (e^{sqrt(x)}-5)/(sqrt(x)) ?
The integral of (e^{sqrt(x)}-5)/(sqrt(x)) is 2(e^{sqrt(x)}-5sqrt(x))+C