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
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
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.
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_"a"^"b" "f"(x) "d"x` = ______
`int_2^4 x/(x^2 + 1) "d"x` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_1^3 x^2*log x "d"x`
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^{pi/2} xsinx dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_0^{pi/2} cos^2x dx` = ______
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_0^9 1/(1 + sqrtx)` dx = ______
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)`
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
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 ______.
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^π(xsinx)/(1 + cos^2x)dx` equals ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is ______.
