00:01
Consider the integral of 2y cubed minus 2y squared plus 1 over y squared minus y dy.
00:07
To evaluate this integral, the first thing we have to do is to apply long division to our integrand.
00:12
Now by long division, 2y cubed minus 2y squared plus 1 divided by y squared minus y will give us 2y as a quotient and then from here we multiply the quotient to the divisor.
00:27
We should get 2y cubed minus 2y squared and then we subtract this from our dividend and from here we should get 1 as a remainder so the final quotient will be 2y plus 1 over y squared minus y.
00:51
That means we can rewrite the integral into the integral of we have 2y plus 1 over y squared minus y and then dy and then from here we separate this into two integrals.
01:05
You should have the integral of 2y dy plus the integral of 1 over y squared minus y dy.
01:14
Now for the second integral, we have to apply partial fraction decomposition and in partial fraction decomposition we need to factor out the denominator.
01:23
That means 1 over y squared minus y will give us 1 over y times y minus 1 and then this will give us a sum of partial fractions with denominators.
01:34
We have y and then we have y minus 1 and since they're both linear then their corresponding numerators are constants...