Advertisements
Advertisements
प्रश्न
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Advertisements
उत्तर
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`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int x^7/(1 + x^4)^2 "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int ("d"x)/(x^3 - 1)`
`int xcos^3x "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
