Advertisements
Advertisements
प्रश्न
Evaluate the following : `int (1)/(1 + x - x^2).dx`
उत्तर
Let I = `int (1)/(1 + x - x^2).dx`
∴ = `I = int1/(1 - (x^2 - x))dx`
∴ = `I = int1/(1-(x^2 - x + 1/4 - (1)/(4)))dx`
∴ = `I = int1/ ((1+1/4) - (x^2 - x + (1/2)^2))dx`
∴ = `I = int 1/ ((sqrt5/2)^2 - (x - 1/2)^2)dx` ...[`int(1/(a^2 - x^2dx) = 1/(2a) log |(a + x)/(a - x)|+c)`]
∴ `I = (1)/(2(sqrt(5)/2))log|(sqrt(5)/(2) + (x - 1/2))/(sqrt(5)/(2) - (x - 1/2))| + c`
∴ `I = (1)/sqrt(5) log |(sqrt(5) - 1 + 2x)/(sqrt(5) + 1 - 2x)|+ c`.
APPEARS IN
संबंधित प्रश्न
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`1/(1 + cot x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
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 ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
`int x^3 e^(x^2) dx`
`int "cosec"^4x dx` = ______.
`int 1/(sin^2x cos^2x)dx` = ______.
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
