Question

Evaluate the following definite integral:

`int_1^2 (3x)/((9x^2 - 1))*dx`

Answer 1

Let I = `int_1^2 (3x)/((9x^2 - 1))*dx`

= `3int_1^2 x/(9x^2 - 1)*dx`

Put 9x² – 1 = t

∴ 18x · dx = dt

∴ x · dx = `(1)/(18)*dx`

When x = 1, t = 9(1)² – 1 = 8

When x = 2, t = 9(2)² – 1 = 35

∴ I = `3int_8^35 (1)/"t"*"dt"/(18)`

= `(1)/(6) int_8^35 "dt"/"t"`

= `(1)/(6)[log|"t"|]_8^35`

= `(1)/(6) (log 35 - log 8)`

∴ I = `(1)/(6)log(35/8)`