PROFESSOR: Calculate the following integral. The definite integral from 1 to 3 of bracket parentheses 2t plus 4 close parentheses i plus open parentheses 3t squared minus 4t close parentheses j close bracket dt. We can find the integral of the i and j component separately. And so this is going to turn out to be two different integrals.

How I'm going to calculate it is the definite integral from 1 to 3 of 2t plus 4 dt. That will give us our i component. And the definite integral from 1 to 3 of the quantity 3t squared minus 4t dt. That will give us our j component.

For the first integral, antiderivative of two t is t squared. And the antiderivative of 4 is 4t. And we will evaluate that from 1 to 3. That would result in 21 minus 5. So i component is 16.

And for the second antiderivative, the antiderivative of 3t squared is t cubed. The antiderivative of minus 4t is minus 2t squared. And we will evaluate that from 1 to 3. Evaluating those, that results in 3 minus negative 1, or 10. So the j component is 10. So the integral from 1 to 3 of 2t plus 4i plus 3t squared minus 4t j dt is equal to 16i plus 10j.