Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Advertisements
उत्तर
Let I = `int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1).dx`
`3x^2 - 2x - 1")"overline(6x^3 + 5x^2 - 7)("2x + 3`
`6x^3 - 4x^2 - 2x`
– + +
`9x^2 + 2x - 7`
`9x^2 - 6x - 3`
– + +
8x – 4
∴ I = `int [(2x + 3) + (8x - 4)/(3x^2 - 2x - 1)] dx`
= `int 2x + 3 + int (8x - 4)/((x - 1)(3x + 1)) dx`
Let `(8x - 4)/((x - 1)(3x + 1)) = "A"/(x - 1) + "B"/(3x + 1)`
∴ 8x – 4 = A(3x + 1) + B(x – 1)
Put x – 1 = 0, i.e. x = 1, we get
8 – 4 = A(4) + B(0)
∴ A = 1
Put 3x + 1 = 0, i.e. x = `-(1)/(3)`, we get
`8(-1/3) - 4 = "A"(0) + "B"(-4/3)`
∴ `(-8 - 12)/(3) = -(4"B")/(3)`
∴ B = 5
∴ `(8x - 4)/((x - 1)(3x + 1)) = (1)/(x - 1) + (5)/(3x + 1)`
∴ `I = 2 int x dx + 3 int 1 dx + int [(1)/(x - 1) + (5)/(3x + 1)] dx`
= `2 (x^2/2) + 3x + int (1)/(x - 1) dx + 5 int (1)/(3x + 1) dx`
= `x^2 + 3x + log |x - 1| + 5/3 log |3x + 1| + c`.
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find :
`∫ sin(x-a)/sin(x+a)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 : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
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)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int sin(logx) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int ("d"x)/(x^3 - 1)`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
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 (dx)/(2 + cos x - sin x)`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Which expression defines a rational function?
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
When is a rational function called improper?
Integration by partial fractions is a method used to integrate which functions?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
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 must be included for a repeated linear factor?
What numerator is used for an irreducible quadratic factor?
