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Question
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
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Solution
Let I = `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Let `2"e"^"x" - 3 = "A" (4"e"^"x" + 1) + "B" "d"/"dx" (4"e"^"x" + 1)`
∴ `2"e"^"x" - 3 = (4"A" + 4"B")"e"^"x" + "A"`
Comparing the coefficients of `"e"^"x"` and constant term on both sides, we get
4A + 4B = 2 and A = - 3
Solving these equations, we get
B = `7/2`
∴ I = `(- 3 (4"e"^"x" + 1) + 7/2(4"e"^"x"))/(4"e"^"x" + 1)` dx
`= - 3 int "dx" + 7/2 int (4"e"^"x")/(4"e"^"x" + 1)`dx
∴ I = `- 3"x" + 7/2 log |4"e"^"x" + 1|` + c ...`[because int ("f" '("x"))/("f"("x")) "dx" = log |"f"("x")| + "c"]`
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