Answer 1

Let I = `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx

Let `2"e"^"x" - 3 = "A" (4"e"^"x" + 1) + "B" "d"/"dx" (4"e"^"x" + 1)`

∴ `2"e"^"x" - 3 = (4"A" + 4"B")"e"^"x" + "A"`

Comparing the coefficients of `"e"^"x"` and constant term on both sides, we get

4A + 4B = 2 and A = - 3

Solving these equations, we get

B = `7/2`

∴ I = `(- 3 (4"e"^"x" + 1) + 7/2(4"e"^"x"))/(4"e"^"x" + 1)` dx

`= - 3 int "dx" + 7/2 int (4"e"^"x")/(4"e"^"x" + 1)`dx

∴ I = `- 3"x" + 7/2 log |4"e"^"x" + 1|` + c    ...`[because int ("f" '("x"))/("f"("x")) "dx" = log |"f"("x")| + "c"]`