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Question
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Solution
Let I = `int (5x^2 - 6x + 3)/(2x − 3)` dx
We perform actual division and express the result as:
`"Dividend"/"Divisor" = "Quotient" + "Remainder"/"Divisor"`
`(5x)/2 + 3/4`
`2x - 3)overline(5x^2 - 6x + 3)`
`- 5x^2 - 15/2x`
(−) (+)
`(3x)/2 + 3`
`- (3x)/2 - 9/4`
(−) (+)
`21/4`
∴ I = `int ((5x)/2 + 3/4 + (21/4)/(2x - 3))` dx
∴ I = `5/2 int x "dx" + 3/4 int "dx" + 21/4 int 1/(2x - 3) "dx"`
∴ I = `5/2 * "x"^2/2 + 3/4"x" + 21/4 * (log |2"x" - 3|)/2 + c`
∴ I = `(5x^2)/4 + (3x)/4 + 21/8 log |2"x" - 3| + c`
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