Advertisements
Advertisements
प्रश्न
Evaluate: `∫8/((x+2)(x^2+4))dx`
Advertisements
उत्तर
Let `I=∫8/((x+2)(x^2+4))dx`
Let `8/((x+2)(x^2+4))=A/(x+2)+(Bx+C)/(x^2+4)`
`8=A(x^2+4)+(Bx+C)(x+2)`
`8=A(x^2+4)+Bx^2+2Bx+Cx+2C`
`8=(A+B)x^2+(2B+c)x+(4A+2C)`
Comparing the coefficients of x2 , x and the constant term, we get
A + B = 0, 2B + C = 0 and 4A + 2C = 8
On solving these equations, we get
A = 1, B = –1, C = 2
`8/((x+2)(x^2+4))=1/(x+2)+(-x+2)/(x^2+4)`
`I=int[1/(x+2)+(-x+2)/(x^2+4)]dx`
`=int1/(x+2)dx-1/2int(2x)/(x^2+4)dx+2int1/(x^2+2^2)dx`
`=log|x+2|-1/2log|x^2+4|+tan^-1(x/2)+c`
`=log|(x+2)/sqrt(x^2+4)|+tan^-1(x/2)+c`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` equals:
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 : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
`int x^2sqrt("a"^2 - x^6) "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 sin(logx) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
`int 1/(4x^2 - 20x + 17) "d"x`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
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`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
