Question

The area of the region bounded by the lines x=1 and y=0 and the curve y=xe^x^2 is...
Answer: (e-1)/2 units^2
Please explain how to get this answer.

Answer 1

It's stating \[\int\limits_{0}^{1} xe^{x^{2}}\,dx = \frac{e - 1}{2}
\]
To verify, perform u-substitution with u = x^2:
\[\int xe^{x^2}\,dx = \int xe^u du = \int \frac{xe^u}{2x}\,=\int \frac{e^u}2 du = \frac12e^u
\]substitute x^2 back in for u and evaluate:\[\left.\frac12e^u\right|_0^1 = \frac12\left(e^1 - e^0 \right) = \frac{e-1}{2}
\]

Answer 2

should say \[\frac12 e^{x^{2}}\] on the left side of that last line, oops. Same answer in this case though.

Answer 3

A Mathematica solution is attached.