# Evaluate the following definite integrals: ∫123x(9x2-1)⋅dx - Mathematics and Statistics

Sum

Evaluate the following definite integrals: 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).

Concept: Fundamental Theorem of Integral Calculus
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