Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{\sqrt{2}} x e^{-\left(x^{2} / 2\right)} d x $$

Short answer

The value of the definite integral is approximately 0.632.

Step-by-step solution

Identify the substitution

To perform the u-substitution, we identify the function within the exponent of 'e'. We let \(u = \frac{x^2}{2}\). Therefore, the differential \(du = x\,dx\).

Perform the substitution

Substitute \(u\) and \(du\) back to the integral function, the integral becomes: \( \int_{0}^{1} e^{-u} du \).

Evaluate the integral

The integral of \(e^{-u}\) is \(-e^{-u}\). We compute \(-e^{-u}\) with the limits of integration from 0 to 1 which results in \(1 - \frac{1}{e}\) or approximately 0.632.

Verify the result

You can use a graphing calculator or an online plotting tool to plot the area under the curve of the function \(xe^{-x^2/2}\) from 0 to \(\sqrt{2}\) to confirm the result.