Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Advertisements
उत्तर
Let I = `int (x^2 + x - 1)/(x^2 + x - 6).dx`
= `int ((x^2 + x - 6) + 5)/(x^2 + x - 6).dx`
= `int [1 + (5)/(x^2 + x - 6)].dx`
= `int 1 dx + 5 int (1)/(x^2 + x - 6).dx`
Let `(1)/(x^2 + x - 6)`
= `(1)/((x + 3)(x - 2)`
= `"A"/(x + 3) + "B"/(x- 2)`
∴ 1 = A(x – 2) + B(x + 3)
Put x 3 = 0, i.e. x = –3, we get
1 = A(– 5) + B(0)
∴ A = `(-1)/(5)`
Put x – 2 = 0, i.e. x = 2, we get
1 = A(0) + B(5)
∴ B = `(1)/(5)`
∴ `(1)/(x^2 + x - 6) = ((-1/5))/(x + 3) + ((1/5))/(x - 2)`
∴ I = `int 1 dx + 5 int [((-1/5))/(x + 3) + ((1/5))/(x - 2)].dx`
= `int 1 dx - int (1)/(x + 3).dx + int (1)/(x - 2).dx`
= x – log|x + 3| + log|x – 2| + c
= `x + log|(x - 2)/(x + 3)| + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
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 : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
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 : `(1)/(sinx*(3 + 2cosx)`
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 with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
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)`
Evaluate: `int 1/("x"("x"^5 + 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.
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x^3tan^(-1)x "d"x`
`int ("d"x)/(x^3 - 1)`
Evaluate `int x log x "d"x`
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
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 (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
`int 1/(x^2 + 1)^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 = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
