What is the integral of (e^{sqrt(x)+3})/(sqrt(x)) ?

The integral of (e^{sqrt(x)+3})/(sqrt(x)) is 2e^{3+sqrt(x)}+C

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integral of (e^{sqrt(x)+3})/(sqrt(x))

\int \frac{e ^{x + 3}}{x} d x

2 e^{3 + x} + C

Solution steps

\int \frac{e ^{x + 3}}{x} d x

\frac{e ^{x + 3}}{x} = e^{3} \frac{e ^{x}}{x}

= \int e^{3} \frac{e ^{x}}{x} d x

Take the constant out:

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

= e^{3} \cdot \int \frac{e ^{x}}{x} d x

Apply u-substitution

: \int 2 e^{u} d u

= e^{3} \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

= e^{3} \cdot 2 \cdot \int e^{u} d u

Use the common integral:

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

= e^{3} \cdot 2 e^{u}

Substitute back

u = x

= e^{3} \cdot 2 e^{x}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{3} e^{x} = e^{3 + x}

= 2 e^{3 + x}

Add a constant to the solution

= 2 e^{3 + x} + C

\int (3 x + 2) (x + 3) d x

\int \frac{cos ( x )}{2 + cos ( x )} d x

\int (x^{3} - x^{2} + 1) d x

\int \frac{e ^{3 x}}{e ^{x} + 1} d x

What is the integral of (e^{sqrt(x)+3})/(sqrt(x)) ?
The integral of (e^{sqrt(x)+3})/(sqrt(x)) is 2e^{3+sqrt(x)}+C