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Question
`int dx/sqrt(9x - 4x^2)` equals:
Options
`1/9 sin^-1 ((9x - 8)/8) + C`
`1/2 sin^-1 ((8x - 9)/9) + C`
`1/3 sin^-1 ((9x - 8)/8) + C`
`1/2 sin^-1 ((9x - 8)/9) + C`
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Solution
`1/2 sin^-1 ((8x - 9)/9) + C`
Explanaton:
Let `I = int dx/sqrt(9x^2 - 4x^2)`
`= 1/2 int dx/sqrt(9/4 x - x^2)`
`= 1/2 int dx/(sqrt(- (x^2 - 9/4 x)))`
`= 1/2 dx/sqrt(- (x^2 - 9/4 + 81/64) + 81/64)`
`= 1/2 int dx/sqrt(81/64 - (x - 9/8)^2) ... [because int 1/(a^2 - x^2) dx = sin^-1 x/a]`
`= 1/2 sin^-1 ((x - 9/8)/(9/8)) +C`
`= 1/2 sin^-1 ((8x - 9)/9) + C`
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