Evaluate the definite integral of the function.
Use a graphing utility to verify your results.

Question

Evaluate the definite integral of the function. ​  ​
Use a graphing utility to verify your results.
​

Quizplus · Accepted Answer

First we apply integration by parts:

$\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C$

Substituting the limits of integration $0$ and $1$ gives:

$\int_0^1 x e^x dx = [xe^x - e^x]_0^1 = (1 - 1/e)$

Using a graphing utility to verify our result, we can plot the function $f(x) = x e^x$ and calculate the area under the curve between $0$ and $1$ using the integral feature of the calculator. The result matches our calculation:

$\int_0^1 x e^x dx \approx 0.6321$

Evaluate the Definite Integral of the Function