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Question
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Solution
Let I = `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Let 2ex + 5 = `"A" (2"e"^x + 1) + "B" "d"/("d"x) (2"e"^x + 1)`
= 2Aex + 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) "d"x`
= `5int "d"x - 4int (2"e"^x)/(2"e"^x + 1) "d"x`
∴ I = 5x – 4log|2e + 1| + c ......`[because int ("f'"(x))/("f"(x)) "d"x = log|"f"(x)| + "c"]`
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