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Question
Evaluate each of the following integral:
Solution
\[I = \int_0^1 x e^{x^2} dx\]
\[ = \frac{1}{2} \int_0^1 e^{x^2} 2xdx\]
Put \[x^2 = z\]
When \[x \to 0, z \to 0\]
When \[x \to 1, z \to 1\]
\[\therefore I = \frac{1}{2} \int_0^1 e^z dz\]
\[ = \frac{1}{2} \left.\times {e^z}\right|_0^1 \]
\[ = \frac{1}{2}\left( e - e^0 \right)\]
\[ = \frac{1}{2}\left( e - 1 \right)\]
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