Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`1/(x^2 - 9)`
Advertisements
उत्तर
Let `1/(x^2 - 9) = 1/((x - 3)(x + 3))`
`= A/(x - 3) + B/(x + 3)`
⇒ 1 ≡ A(x + 3) + B(x - 3)
Put x = 3
1 = A (3 + 3)
⇒ A `= 1/6`
again, put x = -3
1 = B(3 - 3)
⇒ B `= -1/6`
`therefore 1/(x^2 - 9) = 1/6 [1/(x - 3) - 1/(x + 3)]`
`=> int 1/(x^2 - 9) = 1/6 int (1/(x - 3) - 1/(x + 3))` dx
`= 1/6 [log abs (x - 3) - log abs (x + 3)] + C`
`= 1/6 log abs ((x - 3)/(x + 3)) + C`
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int 1/(x(x^3 - 1)) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sec^3x "d"x`
`int ("d"x)/(x^3 - 1)`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
What are the values of \[\mathrm{A}\] and \[\mathrm{B}\] in \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\]?
What numerator should be used for each distinct linear factor in a partial-fraction decomposition?
