Evaluate ∫123x(9x2-1) dx - Mathematics and Statistics

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Sum

Evaluate `int_1^2 (3x)/((9x^2 - 1))  "d"x`

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Solution

Let I = `int_1^2 (3x)/((9x^2 - 1))  "d"x`

= `3int_1^2 x/(9x^2 - 1)  "d"x`

Put 9x2 – 1 = t

∴ 18x dx = dt

∴ x dx = `1/18` dt

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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Chapter 1.6: Definite Integration - Q.4

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