Question Video: Finding the Definite Integration of an Odd Function Using the Properties of Definite Integration

Video Transcript

The function 𝑓 is odd, continuous on the closed interval negative one to seven, and satisfies the definite integral between one and seven of 𝑓 of 𝑥 with respect to 𝑥 equals negative 17. Determine the definite integral between negative one and seven of 𝑓 of 𝑥 with respect to 𝑥.

We’re firstly told that the function 𝑓 is odd. So we recall the following property for integrating odd functions. The definite integral between negative 𝑎 and 𝑎 of 𝑓 of 𝑥 with respect to 𝑥 is equal to zero. We’re also told that the definite integral between one and seven of 𝑓 of 𝑥 with respect to 𝑥 is equal to negative 17. So we split the integral up. And we see that the integral that we’re looking for between negative one and seven of 𝑓 of 𝑥 with respect to 𝑥 is equal to the definite integral between negative one and one of 𝑓 of 𝑥 plus the definite integral between one and seven of 𝑓 of 𝑥.

Now, the function 𝑓 is odd. So by the first property, we see that the definite integral between negative one and one of 𝑓 of 𝑥 with respect to 𝑥 must be equal to zero. Then, we simply take the definite integral between one and seven of 𝑓 of 𝑥 from the question. It’s negative 17. This means the definite integral we’re looking for is equal to zero plus negative 17 which is negative 17.