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Question
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
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Solution
Let I = `int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Let `(3"e")^"2t" + 5 = "A"(4"e"^"2t" - 5) + "B" "d"/"dt" (4"e"^"2t" - 5)`
`= 4 "Ae"^"2t" - 5"A" + "B"(8"e"^"2t")`
∴ `(3"e")^"2t" + 5 = (4"A" + 8"B") "e"^"2t" - 5"A"`
Comparing the coefficients of e2t and constant term on both sides, we get
4A + 8B = 3 and - 5A = 5
Solving these equations, we get
A = - 1 and B = `7/8`
∴ I = `int (- 1(4"e"^"2t" - 5) + 7/8 (8"e"^"2t"))/(4"e"^"2t" - 5)` dt
`= - int "dt" + 7/8 int (8"e"^"2t")/(4"e"^"2t" - 5)` dt
∴ I = `- "t" + 7/8 log |4"e"^"2t" - 5|` + c .....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
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