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

The function 𝑓 is odd, continuous on [βˆ’1, 7], and satisfies ∫_(1) ^(7) 𝑓(π‘₯) dπ‘₯ = βˆ’17. Determine ∫_(βˆ’1) ^(7) 𝑓(π‘₯) dπ‘₯.

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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 π‘₯ with respect to π‘₯ equals negative 17. Determine the definite integral between negative one and seven of 𝑓 of π‘₯ with respect to π‘₯.

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 π‘Ž and π‘Ž of 𝑓 of π‘₯ with respect to π‘₯ is equal to zero. We’re also told that the definite integral between one and seven of 𝑓 of π‘₯ with respect to π‘₯ 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 π‘₯ with respect to π‘₯ is equal to the definite integral between negative one and one of 𝑓 of π‘₯ plus the definite integral between one and seven of 𝑓 of π‘₯.

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 π‘₯ with respect to π‘₯ must be equal to zero. Then, we simply take the definite integral between one and seven of 𝑓 of π‘₯ 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.

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