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:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Integrate the functions:
`1/(1 - tan x)`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : tan5x
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).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 :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int sqrt(1 + sin2x) dx`
`int (sin4x)/(cos 2x) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int(5x + 2)/(3x - 4) dx` = ______
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int ("d"x)/(x(x^4 + 1))` = ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int1/(x^2+4x-5)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
