Discussion
Substituting \( u = x^{2} \) turns \( \displaystyle\int_{0}^{1} x e^{x^{2}}dx \) into
Answer: C
With \( u = x^{2} \) we get \( du = 2x\,dx \), so \( x\,dx = \tfrac{1}{2}du \), and the limits \( x = 0, 1 \) become \( u = 0, 1 \):
\[ \int_{0}^{1} x e^{x^{2}}dx \;=\; \frac{1}{2}\int_{0}^{1} e^{u}du \;=\; \frac{e-1}{2}. \]
The lone factor of \( x \) is exactly what the substitution absorbs, which is why it works here and not for \( \int e^{x^{2}}dx \) alone
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