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

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Question

Evaluate the following.

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

Sum
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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"]`

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Notes

The answer in the textbook is incorrect.

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Chapter 5: Integration - EXERCISE 5.3 [Page 123]

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