Question

Find the antiderivative of (6x)/(x^2 - x - 2)

Answer 1

you must fisrt factor the denominator and use partial fractions.

\[= \int\limits_{}^{}\frac{ 6x }{ (x-2)(x+1) } dx = \int\limits_{}^{} [\frac{ A }{ x-2 } + \frac{ B }{ x+1 }] dx\]

let me know if i should keep going

Answer 2

okay

Answer 3

okay that's enough or okay i should continue?

Answer 4

continue

Answer 5

partial fractions says that: \[\frac{ A }{ x-2 } + \frac{ B }{ x+1 } = \frac{ 6x }{ (x-2)(x+1) }\]

this should come as no shock if you have ever added fractions on paper. you have to make the denominators equal to add them. this is what we will do. we must cross multiply to make the denominators the same:

\[\frac{ A }{ x-2 }\frac{ x+1 }{ x+1 } + \frac{ B }{ x+1 }\frac{ x-2 }{ x-2 } = \frac{ 6x }{ (x-2)(x+1) }\]

\[ =\frac{ Ax + A + Bx - 2B }{ (x-2)(x+1) } = \frac{ 6x }{ (x-2)(x+1) }\]

we can therefore deduce, by comparing numerators of the equivlalent equations, that:
A + B = 6 and
A -2B = 0

solving for A and B: A = 4 and B = 2

you are now dealing with \[\int\limits_{}^{} \frac{ 4 }{ (x-2) } + \frac{ 2 }{ (x+1) } dx\]

Answer 6

okay

Answer 7

which is 4 ln| x-2| + 2 ln | x+1| + C

Answer 8

okay thanks

Answer 9

glad i could help. i hope you learned from this, you will be using this later :)

Answer 10

yes i have :)