Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Advertisements
उत्तर
Let I = `int (1)/(3 + 2sin x - cosx)dx`
Put `tan(x/2)` = t
∴ x = 2 tan–1 t
∴ dx = `(2)/(1 + t^2)dt` and
sinx = `(2t)/(1 + t^2)' cosx = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(3 + 2((2t)/(1 + t^2)) - ((1 - t^2)/(1 + t^2))).(2dt)/(1 + t^2)`
= `int (1 + t^2)/(3(1 + t^2) + 4t - (1 - t^2)).(2dt)/(1 + t^2)`
= `2 int dt/(4t^2 + 4t + 2)`
= `2 int dt/(4t^2 + 4t + 1 + 1)`
= `2 int dt/((2t + 1)^2 + 1^2)`
= `(2)/(2)tan^-1((2t + 1)/1) + c`
= `tan^-1[2tan^-1(x/2) + 1] + c`.
APPEARS IN
संबंधित प्रश्न
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`1/(1 + cot x)`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
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(3x^2 - 8).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
`int (cos2x)/(sin^2x) "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int sec^6 x tan x "d"x` = ______.
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate `int(1 + x + x^2/(2!) )dx`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
