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प्रश्न
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
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उत्तर
Let `I = int (2x + 1)/(x(x - 1)(x - 4)) dx`
Let `(2x + 1)/(x(x - 1)(x - 4)) = A/x + B/(x - 1) + C/(x - 4)`
∴ 2x + 1 = A(x − 1)(x − 4) + Bx(x − 4) + Cx(x − 1) ....(i)
Putting x = 0 in (i), we get
2(0) + 1 = A(0 − 1)(0 − 4) + B(0)(0 − 4) + C(0)(0 − 1)
∴ 2(0) + 1 = A ( −1)( −4) + B0( −4) + 0( −1)
∴ 1 = 4 A
∴ A = `1/4`
Putting x − 1 = 0, i.e., x = 1 in (i), we get
2(1) + 1 = A(0)(−3) + B(1)(1 − 4) + C(1)(0)
∴ 2 + 1 = 0(−3) + B(−3) + 0
∴ 3 = −3B
∴ B = −1
Putting x − 4 = 0, i.e., x = 4 in (i), we get
2(4) + 1 = A(3)(0) + B(4)(0) + C(4)(4 − 1)
∴ 8 + 1 = 0 + 0 + 4C(3)
∴ 9 = C(4)(3)
∴ C = `3/4`
∴ `(2x + 1)/(x(x - 1)(x - 4)) = ((1/4))/x + ((-1))/(x - 1) + ((3/4))/(x - 4)`
∴ I = `int(((1/4))/x + ((-1))/(x - 1) + ((3/4))/(x - 4)) dx`
= `1/4 int 1/x dx - int 1/(x - 1) dx + 3/4 int 1/(x - 4) x`
∴ I = `1/4 log |x| - log |x - 1| + 3/4 log |x - 4| + c`
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