Advertisements
Advertisements
प्रश्न
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
उत्तर
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`
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
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 : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
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 : `(x + 5)/(x^3 + 3x^2 - x - 3)`
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int x^7/(1 + x^4)^2 "d"x`
`int 1/(x(x^3 - 1)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
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 (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
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 (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
