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प्रश्न
Solve the following : `int_2^3 x/(x^2 + 1)*dx`
उत्तर
Let I = `int_2^3 x/(x^2 + 1)*dx`
Put x2 + 1 = t
∴ 2x·dx = dt
∴ x·dx = `"dt"/(2)`
When x = 2, t = 22 + 1 = 5
When x = 3, t = 32 + 1 = 10
∴ I = `int_5^10 (1)/"t"*"dt"/(2)`
= `(1)/(2) int_5^10 "dt"/"t"`
= `(1)/(2)[log |"t"|]_5^10`
= `(1)/(2)(log 10 - log 5)`
= `(1)/(2) log (10/5)`
∴ I = `(1)/(2) log 2`
= `log 2^(1/2)`
= `log sqrt(2)`.
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