What is the value of the integral ∫ e^(x² + ln x) dx ?

(1/2) e^(x²)

e^(x² + x)

x·e^(x²)

None of these

What is the value of the integral:

$\int e^^{x^{2} + l n x} d x$

Recall:

$e^^{a + b} = e^^{a} \cdot e^^{b}$

So,

$e^^{x^^{2} + l n x} = e^^{x^^{2}} \cdot e^^{l n x} = x \cdot e^^{x^^{2}}$

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

Now use substitution:

Let $u = x^2 Rightarrow du = 2x , dx Rightarrow frac{1}{2} du = x , dx$

So,

$int x cdot e^{x^2} , dx = frac{1}{2} int e^u , du = frac{1}{2} e^u + C = frac{1}{2} e^{x^2} + C$

$= 1 \int e^^{u} d u$

$= 1 e^^{u} + C$

$= 1 e^{x^^{2}} + C$