Advertisements
Advertisements
प्रश्न
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Advertisements
उत्तर
Consider, `(2x^2 - 3)/((x^2 - 5)(x^2 + 4))`
Let x2 = m
∴ `(2m - 3)/((m - 5)(m + 4))` ...[proper rational function]
Now, `(2m - 3)/((m - 5)(m + 4)) = A/((m - 5)) + B/((m + 4))`
`(2m - 3)/((m - 5)(m + 4)) = (A(m + 4) + B(m - 5))/((m - 5)(m + 4))`
∴ 2m – 3 = A (m + 4) + B (m – 5)
at m = 5, 2(5) – 3 = A (9) + B (0)
7 = 9A `\implies` A = `7/9`
at m = –4, 2(–4) – 3 = A (0) + B (–9)
–11 = –9B `\implies` B = `11/9`
Thus, `(2m - 3)/((m - 5)(m + 4)) = ((7/9))/((m - 5)) + ((11/9))/((m + 4))` i.e. `(2x^2 - 3)/((x^2 - 5)(x^2 + 4)) = ((7/9))/((x^2 - 5)) + ((11/9))/((x^2 + 4))`
∴ I = `int [((7/9))/(x^2 - 5) + ((11/9))/(x^2 + 4)]dx`
= `7/9 . int 1/(x^2 - (sqrt(5))^2)dx + 11/9 . int 1/(x^2 + (2)^2)dx`
= `7/9 . 1/(2(sqrt(5))) . log [(x - sqrt(5))/(x + sqrt(5))] + 11/9 . 1/2 . tan^-1 (x/2) + c`
∴ I = `7/(18(sqrt(5))) . log [(x - sqrt(5))/(x + sqrt(5))] + 11/18 . tan^-1 (x/2) + c`
∴ `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx = 7/(18(sqrt5)) . log [(x - sqrt(5))/(x + sqrt(5))] + 11/18 . tan^-1 (x/2) + c`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the rational function:
`1/(x(x^4 - 1))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
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)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sec^3x "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int xcos^3x "d"x`
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 the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Which expression defines a rational function?
Which partial-fraction decomposition is used for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})^{2}}\]?
What should be checked before beginning partial-fraction decomposition?
What is done after long division, before writing the appropriate partial-fraction decomposition?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
What is the result of dividing \[x^{2}+1\] by \[x^{2}-5x+6\]?
What must be included for a repeated linear factor?
