Advertisements
Advertisements
प्रश्न
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
उत्तर
Let I = `int sqrt((9 + x)/(9 - x)).dx`
= `int sqrt((9 + x)/(9 - x) xx (9 + x)/(9 + x)).dx`
= `int (9 + x)/sqrt(81 - x^2).dx`
= `int (9)/sqrt(81 - x^2).dx + int x/sqrt(81 - x^2).dx`
= `9 int (1)/sqrt(9^2 - x^2).dx + (1)/(2) int (2x)/sqrt(81 - x^2).dx`
= I1 + I2 ...(Let)
I1 = `9 int (1)/sqrt(9^2 - x^2).dx`
= `9 sin^-1 (x/9) + c_1`
In I2, put 81 – x2 = t
∴ – 2x dx = dt
∴ 2x dx = – dt
I2 = `-(1)/(2) int t^(-1/2) dt`
= `-(1)/(2).t^(1/2)/((1/2)) + c_2`
= `- sqrt(81 - x^2) + c_2`
I = `9 sin^-1 (x/9) - sqrt(81 - x^2) + c`,
where c = c1 + c2 .
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`1/(1 + cot x)`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
`int "dx"/(9"x"^2 + 1)= ______. `
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Evaluate the following integrals : `int cos^2x.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^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int sinx/(sin 3x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*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 :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3)dx`
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate: ∫ |x| dx if x < 0
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int(log(logx))/x "d"x`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int ("d"x)/(x(x^4 + 1))` = ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int 1/(1 + cosα . cosx)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int x^2/sqrt(1 - x^6)dx` = ______.
`int 1/(sin^2x cos^2x)dx` = ______.
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
