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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int 2 (e^{u} + 1) d u

= \int 2 (e^{u} + 1) 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} + 1 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 1 d u)

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

\int 1 d u = u

= 2 (e^{u} + u)

Substitute back

u = x

= 2 (e^{x} + x)

Add a constant to the solution

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

f (x) = \frac{4}{x + 1}, at x = 1

\int \frac{65}{( x - 1 ) ( x ^{2} + 64 )} d x

x \to 25 lim (x - 25)

(x + 1) d y = (x + 6) d x

\frac{\partial}{\partial x} (x^{2} + x)

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