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`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`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 : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
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 : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int sqrt(4^x(4^x + 4)) "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 sqrt((9 + x)/(9 - x)) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int xcos^3x "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
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 ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Integration by partial fractions is a method used to integrate which functions?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
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?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
Which factorisation is correct for \[x^{2}-5x+6\]?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
