Advertisements
Advertisements
प्रश्न
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
पर्याय
`2sqrt(2)`
`2(sqrt(2) + 1)`
2
`2(sqrt(2) - 1)`
Advertisements
उत्तर
`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
संबंधित प्रश्न
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 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`
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
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 : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^pi x*sin x*cos^4x "d"x` = ______.
`int_0^pi x sin^2x dx` = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
`int_a^b f(x)dx` = ______.
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
If f(x) = `(2 - xcosx)/(2 + xcosx)` and g(x) = logex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is ______.
`int_0^1|3x - 1|dx` equals ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` 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 ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate the following definite integral:
`int_4^9 1/sqrt"x" "dx"`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
