Maharashtra State BoardHSC Commerce 12th Board Exam
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Evaluate: ∫x2+x-1x2+x-6) dx - Mathematics and Statistics

Sum

Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx

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Solution

Let I = `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx

`= int(("x"^2 + "x" - 6) + 5)/("x"^2 + "x" - 6)` dx

`= int [("x"^2 + "x" - 6)/("x"^2 + "x" - 6) + 5/("x"^2 + "x" - 6)]` dx

`= int [1 + 5/("x"^2 + "x" - 6)]` dx

`int [1 + 5/(("x + 3")("x - 2"))]` dx

Let `5/(("x + 3")("x - 2")) = "A"/"x + 3" + "B"/"x - 2"`

∴ 5 = A(x - 2) + B(x + 2)   ....(i)

Putting x = 2 in (i), we get

5 = A (0) + B (5)

∴ 5 = 5B

∴ B = 1

Putting x = -3 in (i), we get

5 = A(- 5) + B (0)

∴ 6 = - 5A

∴ A = - 1

∴ `5/(("x + 3")("x - 2")) = (-1)/"x + 3" + 1/"x - 2"`

∴ I = `int [1 + (-1)/"x + 3" + 1/"x - 2"]` dx

`= int "dx" - int 1/"x + 3" "dx" + int1/"x - 2"` dx

∴ I = x - log |x + 3| + log |x - 2| + c

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