Advertisements
Advertisements
Question
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
Options
`2sqrt(2)`
`2(sqrt(2) + 1)`
2
`2(sqrt(2) - 1)`
Advertisements
Solution
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to `2(sqrt(2) - 1)`.
Explanation:
Let I = `int_0^(pi/2) sqrt(1 - sin2x) "d"x`
= `int_0^(pi/2) sqrt((sin^2x + cos^2x - 2 sinx cosx)) "d"x`
= `int_0^(pi/2) sqrt((sinx - cosx)^2) "d"x`
= `int_0^(pi/2) +- (sinx - cosx) "d"x`
= `int_0^(pi/4) - (sin x - cosx) "d"x + int_(pi/4)^(pi/2) (sinx - cosx) "dx`
= `int_0^(pi/4) (cosx - sinx) "d"x + int_(pi/4)^(pi/2) (sinx - cosx) "d"x`
= `[sinx + cosx]_0^(pi/4) + [- cosx - sinx]_(pi/4)^(pi/2)`
= `[(sin pi/4 + cos pi/4) - (sin0 - cos0)] - [(cos pi/2 + sin pi/2) - (cos pi/4 + sin pi/4)]`
= `[(1/sqrt(2) + 1/sqrt(2)) - (+ 1)] - [(0 + 1) - (1/sqrt(2) + 1/sqrt(2))]`
= `(2/sqrt(2) - 1) - (1 - 2/sqrt(2))`
= `2/sqrt(2) - 1 -1 + 2/(sqrt(2))`
= `4/sqrt(2) - 2`
= `2sqrt(2) - 2`
= `2(sqrt(2) - 1)`.
APPEARS IN
RELATED QUESTIONS
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
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^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate : ∫ log (1 + x2) dx
Evaluate `int_0^1 x(1 - x)^5 "d"x`
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_2^3 x/(x^2 - 1)` dx = ______
`int_0^1 (1 - x)^5`dx = ______.
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_0^1 "e"^(5logx) "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^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
