00:01
In this problem, we want to determine whether the following statement is true or false.
00:06
Is the integral of 1 over x to the power of 4, for x ranging between minus 2 to 1, equal to minus 3 over 8? to verify this, let's evaluate our integral.
00:22
This integral is easier to evaluate by rewriting 1 over x to the power of 4 as x to the power of minus 4, and using our exponent rules, we obtain x to the power of minus 3 divided by minus 3, for x ranging between minus 2 to 1, giving us minus 1 over 3x to the power of 3.
00:58
So now we just need to evaluate our integration limits.
01:01
The only problem here is that our function is not defined at x equal to 0, which means that we're going to have to segment this for x ranging between minus 2 to 0, plus the same integral, but for x ranging between 0 to 1.
01:40
And now to evaluate what happens when x equal to 0, we're going to need to invoke limits.
01:48
So this will be equal to the limit when b approaches to 0 of minus 1 over 3x cubed, for x ranging between minus 2 to b, plus the limit when a approaches to 0 of minus 1 over 3x to the power of 3, for x ranging between a to 1.
02:23
But unfortunately, neither of these limits converge...