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
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate : `intlogx/(1+logx)^2dx`
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
Which of the following is true?
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
`int_4^9 1/sqrt(x)dx` = ______.
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 ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
Evaluate `int_-1^1 |x^4 - x|dx`.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
`int_1^2 x logx dx`= ______
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate: `int_-1^1 x^17.cos^4x dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
Which formula expresses \[P_1\] : Reversing Limits?
Under which condition does \[P_6\] give \[\int_{0}^{2a} f(x)\,dx=0\]?
Why can \[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^{2}x\,dx\] be written as \[2\int_{0}^{\frac{\pi}{4}}\sin^{2}x\,dx\]?
