Mathematics

Problem

Let

J=\int_{0}^{1} 2x\,e^{x^{2}}\,dx.

a) Identify an expression

u(x)

that, when used as a new variable, immediately reduces the integrand to a basic exponential form. Explain why this choice is appropriate by relating

\dfrac{du}{dx}

to the factors present in the integrand. b) Rewrite

J

completely in terms of

u

, including new limits of integration. c) Evaluate the rewritten integral to obtain the exact numerical value of

J

. Show every algebraic step clearly.