Perform long division on the integrand, write the proper fraction as a sum of partial fractions, and evaluate the integral:
(integral of) 2y^4dy/y^3 - y^2 + y - 1
After long divison I get:
(integral of)2ydy + 2(integral of)dy + (integral of) 2/y^3 - y^2 + y - 1
I keep getting stuck here. I can't write this as a sum of partial fractions because I can't figure out how to rearrange the denominator so that I CAN write it as a sum o partial fractions. Any ideas?
You can factor
y^3 - y^2 + y - 1
= y^2(y-1) + (y-1)
= (y-1)(y^2 + 1)
To write the integrand as a sum of partial fractions, we need to factorize the denominator, rearrange it, and then equate it to the sum of unknown fractions.
The denominator, y^3 - y^2 + y - 1, cannot be factored easily. In such cases, it is helpful to use polynomial long division to reduce the fraction to a simpler form.
Let's perform polynomial long division on the denominator:
_____________________
y^3 - y^2 + y - 1 | 2y^4 + 0y^3 + 0y^2 + 0y + 0
- (2y^4 - 2y^3 + 2y^2 - 2y)
_____________________
2y^3 + 2y^2 + 2y - 1
Now we can rewrite the integral as:
∫ (2y^4 + 2y^3 + 2y^2 + 2y - 1) / (y^3 - y^2 + y - 1) dy
= ∫ 2y dy + ∫ 2 dy + ∫ (2y^3 + 2y^2 + 2y - 1) / (y^3 - y^2 + y - 1) dy
At this point, we have simplified the integral, and our goal is to express the remaining fraction as a sum of partial fractions. To do this, we need to factorize the denominator.
The denominator, y^3 - y^2 + y - 1, is not easily factorizable. You can check if it has any rational roots by using the Rational Root Theorem or by applying synthetic division with possible integer factors of 1 (the constant term) and checking for remainders of 0. In this case, there are no rational roots.
If the denominator cannot be factored further into linear or quadratic factors, we cannot proceed with partial fraction decomposition using standard methods. It is possible that the integral cannot be expressed in terms of elementary functions.
Therefore, in this case, it seems that further simplification or evaluation of the integral cannot be achieved using standard techniques.