Question

Use the table of integrals to evaluate the integral. 0 11t2e−t dt −1

Answer 1

The integral of
`\( \int_0^(11) t^2e^(-t) \, dt \)` equals
`\( 110e^(-11) - 6e^(-11) - 2 \)`.

Step-by-step explanation:

To solve
`\( \int_0^(11) t^2e^(-t) \, dt \)`, use integration by parts or a table of integrals. Applying integration by parts twice, you'll find
`\( \int t^2e^(-t) \, dt = -t^2e^(-t) + 2\int te^(-t) \, dt \)`. Repeating the process once more leads to
`\( \int te^(-t) \, dt = -te^(-t) + \int e^(-t) \, dt \)`. After integrating, substitute the limits of integration (0 and 11) into the expression to get
`\( 110e^(-11) - 6e^(-11) - 2 \)`.

This approach applies the formula
`\( \int t^ne^(at) \, dt = (t^ne^(at))/(a) - (n)/(a)\int t^(n-1)e^(at) \, dt \)`, using the exponential integral
`\( \int e^(at) \, dt = (e^(at))/(a) \)`. Following the steps of integration by parts simplifies the integral and allows for evaluation with the given limits.

Here is complete question;

"Use the table of integrals to evaluate the following definite integral:

`\[ \int_(0)^(11) t^2e^(-t) \, dt - 1 \]`

Refer to the provided table of integrals to find the necessary antiderivative and apply the Fundamental Theorem of Calculus to determine the value of the expression. Show all intermediate steps in your solution."