# Question Video: Finding the Definite Integration of an Odd Function Using the Properties of Definite Integration Mathematics • Higher Education

The function π is odd, continuous on [β1, 7], and satisfies β«_(1) ^(7) π(π₯) dπ₯ = β17. Determine β«_(β1) ^(7) π(π₯) dπ₯.

01:13

### Video Transcript

The function π is odd, continuous on the closed interval negative one to seven, and satisfies the definite integral between one and seven of π of π₯ with respect to π₯ equals negative 17. Determine the definite integral between negative one and seven of π of π₯ with respect to π₯.

Weβre firstly told that the function π is odd. So we recall the following property for integrating odd functions. The definite integral between negative π and π of π of π₯ with respect to π₯ is equal to zero. Weβre also told that the definite integral between one and seven of π of π₯ with respect to π₯ is equal to negative 17. So we split the integral up. And we see that the integral that weβre looking for between negative one and seven of π of π₯ with respect to π₯ is equal to the definite integral between negative one and one of π of π₯ plus the definite integral between one and seven of π of π₯.

Now, the function π is odd. So by the first property, we see that the definite integral between negative one and one of π of π₯ with respect to π₯ must be equal to zero. Then, we simply take the definite integral between one and seven of π of π₯ from the question. Itβs negative 17. This means the definite integral weβre looking for is equal to zero plus negative 17 which is negative 17.