Answer
$\dfrac{5e^3-5e^2}{2} \approx 31.7412$
Work Step by Step
Given: $I=\int_{0}^{1} 5x e^{x^2+2} \ dx$
Let us consider that $u=x^2+2 \implies dx=\dfrac{du}{2x}$
Now, we have $I= \int_{0}^{1} 5x e^u (\dfrac{du}{2x})$
or, $=\dfrac{5}{2} \int_{0}^{1} e^{u} \ du$
or, $=\dfrac{5}{2} [e^u]_0^1 +C$
or, $=\dfrac{5}{2} [e^{x^2+2}]_0^1 +C$
or, $=\dfrac{5}{2} [e^{(1)^2+2}]-\dfrac{5}{2} [e^{(0)^2+2}]$
or, $=\dfrac{5e^3-5e^2}{2} \approx 31.7412$
