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Question
`int_2^3 dx/(x(x^3 - 1))` = ______.
Options
`(1)/(3) log (208/189)`
`(1)/(3) log (189/208)`
`log (208/189)`
`log (189/208)`
Solution
`int_2^3 dx/(x(x^3 - 1)) = bbunderline((1)/(3) log (208/189))`.
Explanation:
`int_2^3 dx/(x(x^3 - 1))`
x3 − 1 = y
⇒ 3x2 dx = dy
⇒ `x^2 dx = dy/3`
⇒ `int_7^26 (dy/3)/((y + 10 y)`
⇒ `1/3 int_7^26 dy/(y (y + 1))`
`1/y - 1/(y + 1)`
⇒ `(y + 1 - y)/(y(y + 1)`
⇒ `1/3 int_7^26(1/y - 1/(y + 1)) dy`
`int1/(y + a) dy = log (y + a)`
⇒ `1-3 [log y - log (y + 1)]_7^26`
⇒ `1/3 {(log 26 - log 7) - (log 27 - log 8)}`
⇒ `1/3 (log 26/7 - log 27/8)`
= `1/3 log (26 xx 8)/(7 xx 27)`
⇒ `I = 1/3 log 208/189`
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