Advertisements
Advertisements
प्रश्न
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
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 [("x"^2 + "x" − 6)/("x"^2 + "x" − 6) + 5/("x"^2 + "x" − 6)]` dx
`= int [1 + 5/("x"^2 + "x" − 6)]` dx
`int [1 + 5/(("x + 3")("x − 2"))]` dx
Let `5/(("x + 3")("x − 2")) = "A"/"x + 3" + "B"/"x − 2"`
∴ 5 = A(x − 2) + B(x + 3) ....(i)
Putting x = 2 in (i), we get
5 = A (0) + B (5)
∴ 5 = 5B
∴ B = 1
Putting x = − 3 in (i), we get
5 = A(− 5) + B (0)
∴ 5 = − 5A
∴ A = − 1
∴ `5/(("x + 3")("x - 2")) = (-1)/"x + 3" + 1/"x − 2"`
∴ I = `int [1 + (-1)/"x + 3" + 1/"x − 2"]` dx
`= int "dx" - int 1/"x + 3" "dx" + int1/"x − 2"` dx
∴ I = x − log |x + 3| + log |x − 2| + c
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
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
`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_"0"^pi (x"d"x)/(1 + sin 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.
Evaluate: `int (dx)/(2 + cos x - sin x)`
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
