Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Advertisements
उत्तर
Let I = `int (12x + 3)/(6x^2 + 13x - 63).dx`
Let `(12x + 3)/(6x^2 + 13x - 63)`
= `(12x + 3)/((2x + 9)(3x - 7)`
= `"A"/(2x + 9) + "B"/(3x - 7)`
∴ 12 + 3 = A(3x - 7) + B(2x + 9)
Put 2x + 9 = 0, i.e. x = `(-9)/(2)`, we get
`12((-9)/2) + 3 = "A"((-27)/2 - 7)+ "B"(0)`
∴ – 51 = `(-41)/(2)"A"`
∴ A = `(102)/(41)`
Put 3x – 7 = 0, i.x = `(7)/(3)`, we get
`12(7/3) + 3 = "A"(0) + "B"(14/3 + 9)`
∴ 31 = `(41)/(3)"B"`
∴ B = `(93)/(41)`
∴ `(12x + 3)/(6x^2 + 13x - 63)``(12x + 3)/(6x^2 + 13x - 63) = ((102/41))/(2x + 9) + ((93/41))/(3x - 7)`
∴ I = `int [((102/41))/(2x + 9) + ((93/41))/(3x - 7)].dx`
= `(102)/(41) int 1/(2x + 9).dx + 93/41 int 1/(3x - 7).dx`
= `(102)/(41).(log|2x + 9|)/(2) + 93/41.(log|3x - 7|)/(3) + c`
= `(51)/(41)log|2x + 9| + (31)/(41) log|3x - 7| + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
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 : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
`int 1/(x(x^3 - 1)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
