Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

Question

Quizplus · Accepted Answer

Using the Fundamental Theorem of Calculus Part I, we have:

$$\int_1^e xe^{\frac{x^2}{2}}dx = F(e) - F(1)$$
where $F(x) = \int xe^{\frac{x^2}{2}}dx$.

We can find $F(x)$ using substitution method. Let $u = \frac{x^2}{2}$, then $du = xdx$ and we have:

$$F(x) = \int xe^{\frac{x^2}{2}}dx = \int e^u du = e^{\frac{x^2}{2}} + C.$$

Thus, we have:

$$\int_1^e xe^{\frac{x^2}{2}}dx = F(e) - F(1) = e^{\frac{e^2}{2}} - e^{\frac{1}{2}} \approx 20.2422.$$

Therefore, the answer is (D).

Use Part I of the Fundamental Theorem of Calculus to Compute