Question

Evaluate each of the following  integral:

\[\int_0^1 x e^{x^2} dx\]

 

Answer 1

\[I = \int_0^1 x e^{x^2} dx\]
\[ = \frac{1}{2} \int_0^1 e^{x^2} 2xdx\]

Put \[x^2 = z\]

\[\Rightarrow 2x\ dx = dz\]

When \[x \to 0, z \to 0\]

When \[x \to 1, z \to 1\]

\[\therefore I = \frac{1}{2} \int_0^1 e^z dz\]
\[ = \frac{1}{2} \left.\times {e^z}\right|_0^1 \]
\[ = \frac{1}{2}\left( e - e^0 \right)\]
\[ = \frac{1}{2}\left( e - 1 \right)\]