# Evaluate: ∫2x3-3x2-9x+1/2x2-x-10 dx - Mathematics and Statistics

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Sum

Evaluate: int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10) dx

#### Solution

Let I = int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10) dx

We perform actual division and express the result as:

"Dividend"/"Divisor" = "Quotient" + "Remainder"/"Divisor"

x - 1
2"x"^2 - "x" - 10)overline(2"x"^3 - 3"x"^2 - 9"x" + 1)
2"x"^3 - "x"^2 - 10"x"
(-)   (+)  (+)
- 2"x"^2 + "x" + 1
- 2"x"^2 + "x" + 10
(+)  (-)  (-)
- 9

∴ I = int("x - 1" + (-9)/(2"x"^2 - "x" - 10)) dx

= int "x" * "dx" - int 1 * "dx" - 9 int 1/(2"x"^2 - "x" - 10) "dx"

Here 2x2 - x - 10

= 2("x"^2 + 1/2"x" + 1/16 - 5 - 1/16)

= 2 [("x" - 1/4)^2 - 81/16]

∴ I = int "x" * "dx" - int 1 * "dx" - 9/2 int 1/(("x" - 1/4)^2 - (9/4)^2)dx

= "x"^2/2 - "x" - 9/2 * 1/(2 (9/4)) log |("x" - 1/4 - 9/4)/("x" - 1/4 + 9/4)| + "c"_1

= "x"^2/2 - "x" - log |("x" -5/2)/("x + 2")| + "c"_1

= "x"^2/2 - "x" - log|("2x" - 5)/(2("x + 2"))| + "c"_1

= "x"^2/2 - "x" + log|(2("x + 2"))/("2x" - 5)| + "c"_1

= "x"^2/2 - "x" + log |("x + 2")/("2x - 5")| + log 2 + "c"_1

∴ I = "x"^2/2 - "x" + log|("x + 2")/("2x - 5")| + "c"  "where"  "c" = "c"_1 + log 2

Is there an error in this question or solution?
Chapter 5: Integration - MISCELLANEOUS EXERCISE - 5 [Page 139]

#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 5 Integration
MISCELLANEOUS EXERCISE - 5 | Q IV. 5) ii) | Page 139
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