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Question
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
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Solution
Let I = `int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Let 2ex + 5 = A(2ex + 1) + B `"d"/"dx"`(2ex + 1)
= 2 Aex + A + B(2ex )
∴ 2ex + 5 = (2A + 2B)ex + A
Comparing the coefficients of ex and constant term on both sides, we get
2A + 2B = 2 and A = 5
Solving these equations, we get
B = - 4
∴ I = `int (5(2"e"^"x" + 1) - 4(2"e"^"x"))/(2"e"^"x" + 1)`dx
`= 5 int "dx" - 4 int (2"e"^"x")/(2"e"^"x" + 1)`dx
∴ I = 5x - 4 log `|2"e"^"x" + 1|` + c ....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
Notes
The answer in the textbook is incorrect.
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