Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Advertisements
उत्तर
Let I = `int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Resolving into partial fraction, we put
`(2x - 1)/((x - 1)(x + 2)(x - 3)) = "A"/(x - 1) + "B"/(x + 2) + "C"/(x - 3)`
⇒ 2x – 1 = A(x + 2)(x – 3) + B(x – 1)(x – 3) + C(x – 1)(x + 2)
Put x = 1
1 = A(3)(– 2)
⇒ A = `-1/6`
Put x = – 2
– 5 = B(– 3)(– 5)
⇒ B = `- 1/3`
Put x = 3
5 = C(2)(5)
⇒ C = `1/2`
∴ `int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x = - 1/6 int 1/(x - 1) "d"x - 1/3 int 1/(x + 2) "d"x + 1/2 int 1/(x - 3) "d"x`
= `- 1/6 log |x - 1| - 1/3 log|x + 2| + 1/2 log|x - 3| + "C"`
= `- log|x - 1|^(1/6) - log(x + 2)^(1/3) + log(x - 3)^(1/3) + "C"`
Hence, `int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x = log[sqrt(x - 3)/((x - 1)^(1/6) (x + 2)^(1/3))] + "C"`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`1/(x(x^4 - 1))`
`int (xdx)/((x - 1)(x - 2))` equals:
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
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)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` 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.
`int sqrt(4^x(4^x + 4)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int ("d"x)/(x^3 - 1)`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "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 = ______
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` = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?
What should be checked before beginning partial-fraction decomposition?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
Which factorisation is correct for \[x^{2}-5x+6\]?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
