Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
Advertisements
उत्तर
Let `I = int_0^pi (x dx)/ (1 + sin x)`
`I = int_0^pi (pi - x)/ (1 + sin (pi-x)) dx` `...[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
`= int_0^pi (pi - x)/ (1 + sin x) dx`
Adding (i) and (ii), we get
`2 I = int_0^pi (x + pi - x)/ (1 + sin x) dx`
`= pi int_0^pi 1/ (1 + sin x) dx`
`= pi int_0^pi (1 - sin x)/ (1 - sin^2 x) dx`
`= pi int_0^pi (1 - sin x)/(cos^2 x) dx`
`= pi int_0^pi (sec^2 x - tan x sec x) dx`
`= pi [tan x - sec x]_0^pi`
= π [(tan π - sec π) - (tan0 - sec0)]
= π [(0 - (-1)) - (0 - 1)]
= 2π
Hence, I = π
APPEARS IN
संबंधित प्रश्न
Evaluate : `intsec^nxtanxdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_"a"^"b" "f"(x) "d"x` = ______
`int_1^2 1/(2x + 3) dx` = ______
`int_0^{pi/2} xsinx dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_0^{pi/2} cos^2x dx` = ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
`int_0^1|3x - 1|dx` equals ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate: `int_-1^1 x^17.cos^4x dx`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
Which property is represented by \[\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt\]?
If \[f(-x)=-f(x)\], what is \[\int_{-a}^{a} f(x)\,dx\]?
