Advertisements
Advertisements
प्रश्न
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Advertisements
उत्तर
Let I = `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Let `1/((x^2 + 1)(x^2 + 2)) = A/(x^2 + 1) + B/(x^2 + 2)`
⇒ 1 = A(x2 + 2) + B(x2 + 1)
⇒ 1 = (A + B)x2 + (2A + B)
On comparing both sides, we get
A + B = 0 and 2A + B = 0
On solving the above equations, we get
A = 1 and B = –1
∴ I = `int(1/(x^2 + 1) - 1/(x^2 + 2))2xdx`
I = `int (2x)/(x^2 + 1) dx - int (2x)/(x^2 + 2) dx`
I = `log|x^2 + 1| - log|x^2 + 2| + C`
I = `log|(x^2 + 1)/(x^2 + 2)| + C`
APPEARS IN
संबंधित प्रश्न
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x tan-1 x.
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Evaluate:
∫ (log x)2 dx
`int (sinx)/(1 + sin x) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
Evaluate `int 1/(x(x - 1)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`int1/sqrt(x^2 - a^2) dx` = ______
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
`inte^(xloga).e^x dx` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3e^(x^2) dx`
