Question

Solve the following : `int_2^3 x/(x^2 + 1)*dx`

Answer 1

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)`.