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प्रश्न
Solve the following : `int_2^3 x/((x + 2)(x + 3))*dx`
उत्तर
Let I = `int_2^3 x/((x + 2)(x + 3))*dx`
Let `x/((x + 2)x + 3) = "A"/(x + 2) + "B"/(x + 3)` ...(i)
∴ x = A(x + 3) + B(x + 2) ...(ii)
Putting x = – 3 in (ii) we get
– 2 = A
∴ B = 3
Putting x = – 2 in (ii),we get
– 2 = A
∴ A = – 2
From (i), we get
`x/((x + 2(x + 3))) = (-2)/(x + 2) + (3)/(x + 3)`
∴ I = `int_2^3 [(-2)/(x + 2) + 3/(x + 3)]*dx`
= `-2int_2^3 (1)/(x + 2)*dx + 3 int_2^3 (1)/(x + 3)*dx`
= `-2[log|x + 2|]_2^3 + 3[log|x + 3|]_2^3`
= `-2log[log 5 – log 4] + 3[log 6 – log 5]`
= `-2[log(5/4)] + 3[log(6/5)]`
= `3log(6/5) - 2log(5/4)`
= `log(6/5)^2 - 2log(5/4)^2`
= `log(216/125) - log(25/16)`
= `log(216/125 xx 16/25)`
∴ I = `log(3456/3125)`.
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