Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Advertisements
उत्तर
Let I = `int ((x - 1)^2)/(x^2 + 1)^2.dx`
= `int (x^2 - 2x + 1)/(x^2 + 1)^2.dx`
= `int ((x^2 + 1) - 2x)/(x^2 + 1)^2.dx`
= `int [(x^2 + 1)/(x^2 + 1)^2 - (2x)/(x^2 + 1)^2].dx`
= `int (1)/(x^2 + 1)dx - int (2x)/(x^2 + 1)^2.dx`
= I1 – I2 ...(Let)
In I2, Put x2 + 1 = t
∴ 2x dx = dt
= I = `int (1)/(x^2 + 1).dx - int t^-2 dt`
= `tan^-1 x - t^-1/((-1)) + c`
= `tan^-1 x + (1)/(x^2 + 1) + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int "e"^sqrt"x"` dx
`int 1/(xsin^2(logx)) "d"x`
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int (cos x)/(1 - sin x) "dx" =` ______.
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
If f'(x) = 4x3- 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
