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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

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