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
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
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)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int (2x - 7)/sqrt(4x- 1) dx`
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sin(logx) "d"x`
`int x^3tan^(-1)x "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
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
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
`int 1/(x^2 + 1)^2 dx` = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
