How do you integrate #y=(-2-3x)/(x+x^2+2x^3)# using the quotient rule?

Question

Aurora Alvarado · Accepted Answer

#(dy)/(dx)=(-2-4x-15x^2-12x^3)/(x+x^2+2x^3)^2#

#y=(-2-3x)/(x+x^2+2x^3)#

#(dy)/(dx)=((1+2x+6x^2)(-2-3x)-(x+x^2+2x^3)(-3))/(x+x^2+2x^3)^2#

#(dy)/(dx)=(-2-4x-12x^2-3x-6x^2-18x^3+3x+3x^2+6x^3)/(x+x^2+2x^3)^2#

#(dy)/(dx)=(-2-4x-15x^2-12x^3)/(x+x^2+2x^3)^2#

REMEMBER, the quotient rule is

#y=u/v#

#(dy)/(dx)=(v'u-vu')/v^2#

Isaiah Allen · Answer

undefined To integrate $ y = \frac{-2 - 3x}{x + x^2 + 2x^3} $ using the quotient rule, we first express the integrand in partial fractions. After that, we can integrate each term separately. However, the process of finding partial fractions for this expression involves decomposition into simpler fractions. The steps are as follows:

1. Perform long division to see if the degree of the numerator is greater than or equal to the degree of the denominator. If it is, divide the numerator by the denominator.
2. Express the resulting quotient in terms of the divisor plus a proper fraction.
3. Decompose the proper fraction into partial fractions.
4. Integrate each term separately.

Unfortunately, the expression $ y = \frac{-2 - 3x}{x + x^2 + 2x^3} $ is not typically solved using the quotient rule. Instead, it requires partial fraction decomposition followed by integration.