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
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
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/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Find `dy/dx, if y = cos^-1 ( sin 5x)`
`int_0^1 "e"^(2x) "d"x` = ______
Evaluate `int_1^3 x^2*log x "d"x`
`int_2^3 x/(x^2 - 1)` 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_0^1 log(1/x - 1) "dx"` = ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^pi x sin^2x dx` = ______
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "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)` = ______.
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:
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
Evaluate `int_-1^1 |x^4 - x|dx`.
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
