Advertisements
Advertisements
Question
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Advertisements
Solution
Let I = `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Put t = x2
`t/((t + 1)(3t + 4)) = A/(t + 1) + B/(3t + 4)`
t = A(3t + 4) + B(t + 1)
t = (3A + B)t + (4A + B)
On comparing both sides, we get
3A + B = 1 and 4A + B = 0
∴ I = `int (-1)/(x^2 + 1)dx + int (-4)/(3x^2 + 4)dx`
= `-int 1/(x^2 + 1)dx - 4int 1/(3x^2 + 4)dx`
= `-int 1/(x^2 + 1^2)dx - 4int 1/((sqrt(3)x)^2 + 2^2)dx`
= `(-1)/1 tan^-1 (x/2) - 4/(2sqrt(3)) tan^-1 ((sqrt(3)x)/2) + C`
= `-tan^-1x - 2/sqrt(3) tan^-1 ((sqrt(3)x)/2) + C`
RELATED QUESTIONS
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int sqrt(4^x(4^x + 4)) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x sin2x cos5x "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
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 \[x^{2}-5x+6\]?
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 \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\]?
What must be included for a repeated linear factor?
