Advertisements
Advertisements
Question
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Advertisements
Solution
Let `(2x - 3)/((x^2 - 1)(2x + 3))`
`= (2x - 3)/((x - 1)(x + 1) (2x + 3))`
`= A/(x - 1) + B/(x + 1) + C/(2x + 3)`
⇒ 2x - 3 = A(x + 1)(2x + 3) + B(x - 1)(2x + 3) + C(x - 1)(x + 1) .... (1)
Putting x = 1 in equation (1),
2(1) - 3 = A(1 + 1)(2 + 3)
⇒ -1 = A (2) (5)
⇒ A `= -1/10`
Putting x = -1 in equation (1),
-2 -3 = B (-1 -1)(-2 + 3)
⇒ -5 = B (-2)(1)
⇒ B `= 5/2`
Putting `x = -3/2` in equation (1),
-3 -3 = C `(-3/2 -1)(-3/2 + 1)`
⇒ -6 = C `(-5/2)(-1/2)`
⇒ C =`- 6 xx 4/5 = -24/5`
`therefore (2x - 3)/((x^2 - 1)(2x + 3)) = - 1/(10(x - 1)) + 5/(2(x + 1)) - 24/(5(2x + 3))`
`therefore int (2x - 3)/((x^2 - 1)(2x+ 3)) dx = -1/10 int 1/(x - 1) dx + 5/2 int 1/(x + 1) dx -24/5 int 1/(2x + 3) dx`
` = - 1/10 log (x - 1) + 5/2 log (x + 1) - 24/5 log ((2x + 3)/2) + C`
`= 5/2 log (x + 1) - 1/10 log (x - 1) - 12/5 log (2x+ 3) + C`
APPEARS IN
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`1/(x(x^4 - 1))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
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.
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 (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int x sin2x cos5x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
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 (dx)/(2 + cos x - sin x)`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
