All type of definite integral questions and answers
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Evaluate each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
This is the only indefinite integral in this section and by now we should be getting pretty good with these so we won’t spend a lot of time on this part. This is here only to make sure that we understand the difference between an indefinite and a definite integral. The integral is,
[Return to Problems]
(b)
Recall from our first example above that all we really need here is any anti-derivative of the integrand. We just computed the most general anti-derivative in the first part so we can use that if we want to. However, recall that as we noted above any constants we tack on will just cancel in the long run and so we’ll use the answer from (a) without the “+c”.
Here’s the integral,
Remember that the evaluation is always done in the order of evaluation at the upper limit minus evaluation at the lower limit. Also be very careful with minus signs and parenthesis. It’s very easy to forget them or mishandle them and get the wrong answer.
Notice as well that, in order to help with the evaluation, we rewrote the indefinite integral a little. In particular we got rid of the negative exponent on the second term. It’s generally easier to evaluate the term with positive exponents.
[Return to Problems]
(c)
This integral is here to make a point. Recall that in order for us to do an integral the integrand must be continuous in the range of the limits. In this case the second term will have division by zero at and since is in the interval of integration, i.e. it is between the lower and upper limit, this integrand is not continuous in the interval of integration and so we can’t do this integral.
Note that this problem will not prevent us from doing the integral in (b) since is not in the interval of integration.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
This is the only indefinite integral in this section and by now we should be getting pretty good with these so we won’t spend a lot of time on this part. This is here only to make sure that we understand the difference between an indefinite and a definite integral. The integral is,
[Return to Problems]
(b)
Recall from our first example above that all we really need here is any anti-derivative of the integrand. We just computed the most general anti-derivative in the first part so we can use that if we want to. However, recall that as we noted above any constants we tack on will just cancel in the long run and so we’ll use the answer from (a) without the “+c”.
Here’s the integral,
Remember that the evaluation is always done in the order of evaluation at the upper limit minus evaluation at the lower limit. Also be very careful with minus signs and parenthesis. It’s very easy to forget them or mishandle them and get the wrong answer.
Notice as well that, in order to help with the evaluation, we rewrote the indefinite integral a little. In particular we got rid of the negative exponent on the second term. It’s generally easier to evaluate the term with positive exponents.
[Return to Problems]
(c)
This integral is here to make a point. Recall that in order for us to do an integral the integrand must be continuous in the range of the limits. In this case the second term will have division by zero at and since is in the interval of integration, i.e. it is between the lower and upper limit, this integrand is not continuous in the interval of integration and so we can’t do this integral.
Note that this problem will not prevent us from doing the integral in (b) since is not in the interval of integration.
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