Indefinite integral

Question

∫(5x+1)/(x-4)(x+1)² dx

Answer 1

Partial Fraction Decomposition:

(5x+1)/[(x-4)(x+1)²] = A/(x-4) + B/(x+1) + C/(x+1)²

Multiply both sides by (x-4)(x+1)²:

5x+1 = A(x+1)² + B(x-4)(x+1) + C(x-4)

At this point you could let x = -1, to find the value of C, then x = 4 to find the value of A. Once you have those values you can choose any value for x to determine B. You can also equate coefficients to find the values of A,B,C (resulting in 3 equations an 3 unknowns).

x = -1:

-4 = -5C

C=4/5

x = 4:

21 = 25A

A = 21/25

x = 0:

1 = 21/25 + (-4B) + (-4)(4/5)

4B = 21/25 - 16/5 - 1

4B = 21/25 - 80/25 - 25/25

4B = -84/25

B = -21/25

So the original integral can be rewritten as:

∫ [(21/25)/(x - 4) + (-21/25)/(x+1) + (4/5)/(x+1)²]dx

Antiderivative:

(21/25)ln(|x - 4|) - (21/25)ln(|x+1|) - 4/[5(x+1)] + C , using u-sub for the last term:(4/5) ∫ u^-2du.

The following graph depicts the original function, the partial fraction version, and shows that the derivative of the antiderivative above matches the original.

desmos.com/calculator/w5okn8xeil