Question

Find the partial fractions decomposition and an antiderivative.

Answer 1

\[\frac{ x^{?} + x + 2 }{ x^{2} + 2x - 8 } = \frac{ A }{ (x-2) } + \frac{ B }{ (x + 4) }\]

Answer 2

implied that by cross multiplying the right side: A(x+4) + B(x-2) = numerator of other side. you can solve for A and B

Answer 3

The antiderivative will be A*ln(x-2) + B*ln(x+4) + C.

Answer 4

okay

Answer 5

so i'm cross multiplying a/x - 2 and b/x + 4 ?

Answer 6

well basically, you're adding them. and to add fractions that don't have the same denominator, you need to cross multiply by what the fractions need to have the same denominators and multiply the denominators

\[\frac{ a }{ b } + \frac{ c }{ d } = \frac{ ad + bc }{ }\]

and we want to match both sides

Answer 7

\[\frac{ a }{ b } + \frac{ c }{ d } = \frac{ ad + bc }{ bd }\]

Answer 8

so A/x + 4) + (x - 2)B over (x - 2)(x + 4) ?

Answer 9

so?

Answer 10

yes

Answer 11

okay so what do i do next? :/

Answer 12

@Snipa420

Answer 13

Basically A(x+4) + B(x-2) = x^2 + x +2

Answer 14

ok

Answer 15

Plug in -4 for x to find B and 2 for x to find A

Answer 16

into x^2 + x + 2 right?

Answer 17

so whats next?

Answer 18

so my answer will be 8/x - 2 + -18/x + 1 right?

Answer 19

i meant 8/x - 2 + -18/x + 4

Answer 20

@Snipa420