Question

How do you evaluate the definite integral `int (t^3-9t)dt` from [-1,1]?

Answer 1

`0.`

Explanation:

Observe that the Integrand `f(t)=t^3-9t, t in [-1,1],` is an odd
continuous function.

So, at a moment's glance, we can answer `0`, if we know the following Result :

`int_-a^af(t)dt=0, if f : [-a,a] to RR" is continuous & odd on "[-a,a].`

Alternatively, `int_-1^1(t^3-9t)dt=[t^4/4-(9t^2)/2]_-1^1`

`=[(1/4-9/2)-(1/4-9/2)]=0.`

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