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 : `intlogx/(1+logx)^2dx`
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
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 following integral:
`int_0^1 x(1 - x)^5 *dx`
`int_0^2 e^x dx` = ______.
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^1 (1 - x)^5`dx = ______.
`int_0^1 "e"^(5logx) "d"x` = ______.
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
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 ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
