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Question
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))*dx`
Solution
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)`
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