Advertisements
Advertisements
प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Advertisements
उत्तर
`int x^2/(x^4+x^2-2)dx`
`=int x^2/((x^2-1)(x^2+2))dx`
`=int x^2/((x+1)(x-1)(x^2+2))dx`
Using partial fraction
`x^/((x+1)(x-1)(x^2+2))=A/(x-1)+B/(x+1)+(Cx+D)/(x^2+2)`
`=(A(x+1)(x^2+2)+B(x-1)(x^2+2)+(Cx+D)(x+1)(x-1))/((x+1)(x-1)(x^2+2))`
Equating the coefficients from both the numerators we get,
A + B + C = 0........(1)
A - B + D = 1........(2)
2A + 2B - C = 0........(3)
2A - 2B - D= 0........(4)
Solving the above equations we get,
`A=1/6, B=-1/6, C=0, D=2/3`
Our Integral becomes
`intx^/((x+1)(x-1)(x^2+2))dx=1/(6(x-1))-1/(6(x+1))+2/(3(x^2+2))dx`
`=1/6log(x-1)-1/6log(x+1)+2/3xx1/sqrt2 tan^-1 (x/sqrt2)+C`
`=1/6[log(x-1)-log(x+1)+2sqrt2tan^-1 (x/sqrt2)]+C`
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` 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 : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(x^3 - 1)`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
Evaluate `int x^2"e"^(4x) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
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 (dx)/(2 + cos x - sin x)`
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)
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 ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
