Advertisements
Advertisements
Question
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Advertisements
Solution
Let I = `int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Let `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
= `"A"/(x^2 + 1) + "b"/(x^2 - 2) + "c"/(x^2 + 3)`
∴ x2 = A(x2 − 2)(x2 + 3) + B(x2 + 1)(x2 + 3) + C(x2 + 1)(x2 − 2) ........(i)
Putting x2 = 2 in (i), we get
2 = B × 3 × 5
∴ B = `2/15`
Putting x2 = −3 in (i), we get
−3 = C × (– 2) × (– 5)
∴ C = `(-3)/10`
Putting x2 = −1 in (i), we get
−1 = A × (–3) × 2
∴ A = `1/6`
∴ `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) = (1/6)/(x^2 + 1) + (2/15)/(x^2 - 2) + ((-3)/10)/(x^2 + 3)`
∴ I = `int[1/(6(x^2 + 1)) + 2/(15(x^2 - 2)) - 3/(10(x^2 + 3))] "d"x`
= `1/6 int 1/(x^2 + 1) "d"x + 2/15 int 1/(x^2 - 2) "d"x - 3/10 int 1/(x^2 + 3) "d"x`
= `1/6 int 1/(x^2 + 1) "d"x + 2/15 int 1/(x^2 - (sqrt(2))^2) "d"x - 3/10 int 1/(x^2 + (sqrt(3))^2) "d"x`
= `1/6 tan^-1x + 2/15 xx 1/(2 xx sqrt(2)) log|(x - sqrt(2))/(x + sqrt(2))| - 3/10 xx 1/sqrt(3) tan^-1 (x/sqrt(3)) + "c"`
∴ I = `1/6 tan^-1x + 1/(15sqrt(2)) log|(x - sqrt(2))/(x + sqrt(2))| - sqrt(3)/10 tan^-1 (x/sqrt(3)) + "c"`
APPEARS IN
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`1/(x^4 - 1)`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int sec^3x "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int xcos^3x "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 log x "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
Evaluate: `int (dx)/(2 + cos x - sin x)`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
`int 1/(x^2 + 1)^2 dx` = ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
Which expression defines a rational function?
Which partial-fraction decomposition is used for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})^{2}}\]?
Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?
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?
For \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\], which equation results after clearing denominators?
What are the values of \[\mathrm{A}\] and \[\mathrm{B}\] in \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\]?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
What must be included for a repeated linear factor?
What numerator is used for an irreducible quadratic factor?
