Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`x/((x^2+1)(x - 1))`
उत्तर
Let `x/((x^2 + 1)(x - 1)) = (Ax + B)/(x^2 + 1) + C/((x - 1))`
⇒ x = (A) (+ B)(x - 1) + C = `1/2`
Put x = 1
1 = 0 + 2C
⇒ C `= 1/2`
On comparing the coefficients of x2 or x
0 = A + C
⇒ A = `- 1/2`
and 1 = - A + B
⇒ B `= 1/2`
Hence, `int x/((x^2 + 1)(x - 1)) dx`
`= int (- 1/2 x + 1/2)/(x^2 + 1) dx + 1/2 int 1/(x - 1) dx`
`= -1/2 int (x - 1)/(x^2 + 1) dx + 1/2 log abs (x - 1) + C`
`= 1/4 int (2x)/(x^2 + 1) + 1/2 int 1/(x^2 + 1) dx + 1/2 log abs (x - 1) + C`
`= - 1/4 log abs (x^2 + 1) + 1/2 tan^-1 x + 1/2 log abs (x - 1) + C`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
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 : `(12x + 3)/(6x^2 + 13x - 63)`
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)`
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 w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int 1/(x(x^3 - 1)) "d"x`
`int ("d"x)/(x^3 - 1)`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int xcos^3x "d"x`
`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 = ?
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 = ______
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
Evaluate `int x^2"e"^(4x) "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`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
`int 1/(x^2 + 1)^2 dx` = ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
