Advertisements
Advertisements
प्रश्न
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Advertisements
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
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
`int (dx)/(sin^2 x cos^2 x)` equals:
Write a value of
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).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 - 2cos 2x).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
`int logx/(log ex)^2*dx` = ______.
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
`int sqrt(1 + "x"^2) "dx"` =
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
Evaluate: `int log ("x"^2 + "x")` dx
Evaluate: `int "e"^sqrt"x"` dx
`int 1/(cos x - sin x)` dx = _______________
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int(log(logx) + 1/(logx)^2)dx` = ______.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int (1+x+x^2/(2!))dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`intx sqrt(1 +x^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).
What is the value of \[\int\sin^3x\cos^2x\,dx\]?
