Advertisements
Advertisements
प्रश्न
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Advertisements
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 (sinx)/sqrt(36-cos^2x)dx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `int sqrt(1 + sin 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^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following : `int (logx)2.dx`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int 1/(xsin^2(logx)) "d"x`
`int cos^7 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"`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate `int1/(x(x - 1))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
