Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Advertisements
उत्तर
Let I = `int (x^2 + 2)/((x - 1)(x + 2)(x + 3)).dx`
Let `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
= `"A"/(x - 1) + "B"/(x + 2) + "C"/(x + 3)`
∴ x2 + 2 = A(x + 2)(x + 3) + B(x – 1)(x + 3) + C(x – 1)(x + 2)
Put x – 1 = 0, i.e. x = 1, we get
1 + 2 = A(3)(4) + B(0)(4) + C(0)(3)
∴ 3 = 12A
∴ A = `(1)/(4)`
Put x + 2 = 0, i.e. x = – 2, we get
4 + 2 = A(0)(1) + B(– 3)(1) + C(– 3)(0)
∴ 6 = – 3B
∴ B = – 2
Put x + 3 = 0, i.e. x = – 3we get
9 + 2 = A(– 1)(0) + B(– 4)(0) + C(– 4)(– 1)
∴ 11 = 4C
∴ C = `(11)/(4)`
∴ `(x^2 + 2)/((x - 1)(x + 2)(x + 3)) = ((1/4))/(x - 1) + (-2)/(x + 2) + ((11/4))/(x + 3)`
∴ I = `int [((1/4))/(x - 1) + (-2)/(x + 2) + ((11/4))/(x + 3)].dx`
= `(1)/(4) int (1)/(x - 1).dx - 2 int(1)/(x + 2).dx + (11)/(4) int (1)/(x + 3).dx`
= `(1)/(4)log|x - 1| - 2 log|x + 2| + (11)/(4)log | x + 3| + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
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 following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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 : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`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 sqrt((9 + x)/(9 - x)) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int xcos^3x "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 = ?
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int x log x "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
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
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)`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
