Advertisements
Advertisements
Question
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Advertisements
Solution
Since `(1-x^2)/(x (1 - 2x)) = (1 - x^2)/(x - 2x^2)` is an improper fraction, therefore we convert it into a peoper fraction by long division method, we get
`(x^2 - 1)/(2x^2 - x) = 1/2 + (x/2 - 1)/(2x^2 - x)`
`= int (-1 + x^2)/(-x + 2x^2) dx`
`= 1/2 int dx 1/2 int (x-2)/(2x^2 - x) dx`
Now, `(x - 2)/(2x^2 - x) = (x - 2)/(x (2x - 1))`
`= A/x + B/(2x - 1)`
⇒ x - 2 = A (2x - 1) + Bx ......(i)
Putting x = 0 in (i), we get
-2 = A (-1)
⇒ A = 2
Putting `x = 1/2` in (i), we get
`1/2 -2= B (1/2)`
⇒ 1 - 4 = B
⇒ B = -3
∴ `(x - 2)/ (2x^2 - x) = 2/x - 3/ (2x - 1) = 2/x + 3/ (1 - 2x)`
We have,
`int (1 - x^2)/(x (1 - 2x)) dx`
`= 1/2 int 1 dx + 1/2 int (2/x + 3 /(1 - 2x)) dx`
`= 1/2x + log |x| -3/4 log |1 - 2x| + C`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
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 : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sin(logx) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x sin2x cos5x "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
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`
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
For \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\], which equation results after clearing denominators?
