Evaluate ∫2ex+52ex+1 dx - Mathematics and Statistics

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Sum

Evaluate `int (2"e"^x + 5)/(2"e"^x + 1)  "d"x`

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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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Chapter 1.5: Integration - Q.4

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