Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Advertisements
उत्तर
Let I = `int (1)/(2 + cosx - sinx).dx`
Put `tan (x/2)` = t
∴ x 2 tan–1 t
∴ dx = `(2dt)/(1 + t^2) and sin x = (2t)/(1 + t^2), cosx = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(2 + ((1 - t^2)/(1 + t^2)) - ((2t)/(1 + t^2))).(2dt)/(1 + t^2)`
= `int (1 + t^2)/(2 + 2t^2 + 1 - t^2 - 2t).(2dt)/(1 + t^2)`
= `2 int (1)/(t^2 - 2t + 3)dt`
= `2 int (1)/((t^2 - 2t + 1) + 2)dt`
= `2 int (1)/((t - 1)^2 + (sqrt(2))^2).dt`
= `2 xx (1)/sqrt(2)tan^-1 ((t - 1)/sqrt(2)) + c`
= `sqrt(2)tan^-1[(tan(x/2) - 1)/sqrt(2)] + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Evaluate: `int 1/(x(x-1)) dx`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(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 : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
`int logx/(log ex)^2*dx` = ______.
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int x^2/sqrt(1 - x^6)` dx = ________________
`int x^x (1 + logx) "d"x`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int sin^-1 x`dx = ?
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate:
`int sqrt((a - x)/x) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
