Advertisements
Advertisements
प्रश्न
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
विकल्प
`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 e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]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^a sqrtx/(sqrtx + sqrt(a-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.
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
`int_1^2 1/(2x + 3) dx` = ______
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
If `int_0^"a" sqrt("a - x"/x) "dx" = "K"/2`, then K = ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
`int_0^1 1/(2x + 5) dx` = ______.
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
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 ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate:
`int_0^1 |2x + 1|dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
