3. evaluate ∫₁² xeˣ² dx

Explanation:

Step1: Use substitution

Let $u = x^{2}$, then $du=2xdx$, and $xdx=\frac{1}{2}du$. When $x = 1$, $u = 1$; when $x = 2$, $u = 4$.

Step2: Rewrite the integral

$\int_{1}^{2}xe^{x^{2}}dx=\frac{1}{2}\int_{1}^{4}e^{u}du$

Step3: Integrate $e^{u}$

The antiderivative of $e^{u}$ is $e^{u}$. So $\frac{1}{2}\int_{1}^{4}e^{u}du=\frac{1}{2}[e^{u}]_{1}^{4}$

Step4: Evaluate the definite - integral

$\frac{1}{2}(e^{4}-e^{1})=\frac{e^{4}-e}{2}$

Answer:

$\frac{e^{4}-e}{2}$