# Evaluate: ∫2ex-34ex+1 dx - Mathematics and Statistics

Sum

Evaluate: int (2"e"^"x" - 3)/(4"e"^"x" + 1) dx

#### Solution

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"]

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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 5 Integration
Miscellaneous Exercise 5 | Q 4.2 | Page 138