Advertisements
Advertisements
प्रश्न
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Advertisements
उत्तर
Let I = `int_0^(2π) (1)/(1 + e^(sin x)`dx ...(i)
Applying property,
`int_0^af(x)dx = int_0^af(a-x)dx,` we get
I = `int_0^(2pi) dx/(1+e^(sin(2pi-x)))`
= `int_0^(2pi)dx/(1+e^(-sinx))`
= `int_0^(2pi)dx/(1+1/e^(sinx))`
= `int_0^(2pi)(e^(sinx)dx)/(e^(sinx)+1)` ...(ii)
On adding equations (i) and (ii), we get
2I = `int_0^(2pi)dx/(1+e^(sinx))+int_0^(2pi)(e^(sinx)dx)/(1+e^(sinx))`
= `int_0^(2pi)((1+e^(sinx))/(1+e^(sinx)))dx`
= `int_0^(2pi)1.dx`
⇒ 2I = `[x]_0^(2pi)`
⇒ 2I = [2π]
⇒ I = π
संबंधित प्रश्न
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate : `intsec^nxtanxdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_-9^9 x^3/(4 - x^2)` dx = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0
⇒ `1/4 (square - square)` = 0
⇒ b4 – `square` = 0
⇒ (b2 – a2)(`square` + `square`) = 0
⇒ b2 – `square` = 0 as a2 + b2 ≠ 0
⇒ b = ± `square`
If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
`int_0^1|3x - 1|dx` equals ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Solve.
`int_0^1e^(x^2)x^3dx`
