Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
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`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`(1+ log x)^2/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of
Write a value of
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals : `int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x:
`(10x^9 10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : `(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following : `int sinx/(sin 3x).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` 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 (5"x" + 1)^(4/9)` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int sqrt(1 + sin2x) "d"x`
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int cos^7 x "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate `int 1/(x(x-1))dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
