Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
उत्तर
Let `x/((x - 1)^2(x + 2))`
`= A/((x - 1)) = B/((x - 1)^2) + C/((x + 2))`
⇒ x = A(x - 1)(x + 2) + B(x + 2) + C(x - 1)2 ...(1)
Put x = 1
1 = 3B
⇒ B = `1/3`
Put x = -2
-2 = C (-2 - 1)2
⇒ C = `(-2)/9`
On comparing the coefficients of x2
A = `-C = 2/9`
Hence, `int x/((x+ 1)^2(x - 2))` dx
`= int 2/ (9 (x - 1)) dx + int 1/ (3 (x - 1)^2) dx - int 2/ (9(x + 2)) dx`
`= 2/9 log abs (x - 1) + 1/3 int (x - 1)^-1/-1 - 2/9 log abs (x + 2) + C`
`= 2/9 log abs ((x - 1)/(x + 2)) - 1/(3(x + 1)) + C`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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)/(sinx*(3 + 2cosx)`
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 : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int (2"x" + 1)/("x"("x - 1")("x - 4"))` dx
Evaluate: `int "x"/(("x - 1")^2("x + 2"))` dx
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` 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`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int x sin2x cos5x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "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 (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"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_-2^1 sqrt(5 - 4x - x^2)dx`
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
