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Question
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
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Solution
Let I = `int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Let 20 - 12ex = A(3ex - 4) + B `"d"/"dx"`(3ex - 4)
= 3 Aex - 4A + 3Bex
∴ 20 - 12ex = (3A + 3B)ex - 4A
Comparing the coefficients of ex and constant term on both sides, we get
- 4A = 20 and 3A + 3B = - 12
Solving these equations, we get
A = -5 and B = 1
∴ I = `int (-5(3"e"^"x" - 4) + 3"e"^"x")/(3"e"^"x" - 4)`dx
`= - 5 int "dx" + int (3"e"^"x")/(3"e"^"x" - 4)` dx
∴ I = - 5x + log `|(3"e"^"x" - 4)|` + c ....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
Notes
The answer in the textbook is incorrect.
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