Advertisements
Advertisements
Question
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Advertisements
Solution
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
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(3x -1)/(x + 2)^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:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`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 : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
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]`
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)/(2cosx + 3sinx)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sin(logx) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "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`
Evaluate `int x^2"e"^(4x) "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^4 - x^2 - 12)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
What numerator should be used for each distinct linear factor in a partial-fraction decomposition?
