Find the antiderivative of the function (-x^2+4x+10)/(x+2)(x+1)^2

To determine the antiderivative of the function, we'll
have to calculate the indefinite integral. To ease the work of finding the primitive,
we'll decompose into partial fractions the given
ratio.

We'll apply Heaviside's
method:

(-x^2+4x+10)/(x+2)(x+1)^2 = A/(x+2) + B/(x+1) +
C/(x+1)^2

A = [-(-2)^2 + 4*(-2) +
10]/(-2+1)^2

A = -2/1 = -2

C =
[-(-1)^2 + 4*(-1) + 10]/(-1+2)

C =
5

To determine B, we'll re-write the
identity:

(-x^2+4x+10)/(x+2)(x+1)^2 = -2/(x+2) + B/(x+1) +
5/(x+1)^2

-x^2+4x+10 = -2(x+1)^2 + B(x+1)(x+2) +
5(x+2)

We'll replace x by 0 and we'll
get:

10 = -2 + 2B + 10

2B = 2
=> B = 1

The complete decomposition in partial
fractions is:

(-x^2+4x+10)/(x+2)(x+1)^2 = -2/(x+2) +
1/(x+1) + 5/(x+1)^2

Now, we'll integrate both
sides:

Int (-x^2+4x+10) dx/(x+2)(x+1)^2 = Int -2dx/(x+2) +
Int dx/(x+1) + Int 5dx/(x+1)^2

Int -2dx/(x+2) = -2ln|(x+2)|
+ C = ln [1/(x+2)^2] + C

Int dx/(x+1) = ln|x+1| +
C

Int 5dx/(x+1)^2 = -5/(x+1) +
C

The antiderivative of the given function
is: F(x) = ln [1/(x+2)^2] + ln|x+1| - 5/(x+1) + C