Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Advertisements
उत्तर
Let I = `int (1)/(2sin 2x - 3)dx`
Put tan x = t
∴ x = tan–1 t
∴ dx = `dt/(1 + t^2) and sin 2x = (2t)/(1 + t^2)`
∴ I = `int(1)/(2((2t)/(1 + t^2)) - 3).dt/(1 + t^2)`
= `int (1 + t^2)/(4t - 3 - 3t^2).dt/(1 + t^2)`
= `int (1)/(-3t^2 + 4t - 3)dt`
= `(1)/(3) int (1)/(t^2 - 4/3t + 1)dt`
= `-(1)/(3) int (1)/((t^2 - 4/3t + 4/9) - (4)/(9) + 1)dt`
= `-(1)/(3) int (1)/((t - 2/3)^2 + (sqrt(5)/3)^2)dt`
= `-(1)/(3) xx (1)/((sqrt(5)/3))tan^-1 ((t - 2/3)/(sqrt(5)/3)) + c`
= `-(1)/sqrt(5)tan^-1 ((3t - 2)/sqrt(5)) + c`
= `-(1)/sqrt(5)tan^-1((3tan x - 2)/(sqrt(5))) + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of
Write a value of
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals:
tan2x dx
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
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)/(cosx - sinx).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int logx/x "d"x`
`int x^x (1 + logx) "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int1/(4 + 3cos^2x)dx` = ______
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate `int 1/("x"("x" - 1)) "dx"`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate `int 1/(x(x-1))dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) 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).
Evaluate:
`intsqrt(sec x/2 - 1)dx`
