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
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| 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_0^4 |x - 1| dx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
`int_0^1 "e"^(2x) "d"x` = ______
`int_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
`int_0^{pi/2} log(tanx)dx` = ______
`int_2^3 x/(x^2 - 1)` dx = ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
Which of the following is true?
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
`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:
`int (dx)/(e^x + e^(-x))` is equal to ______.
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
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 ______.
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`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
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 - 5)^5`dx
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Under which condition does \[P_6\] give \[\int_{0}^{2a} f(x)\,dx=2\int_{0}^{a} f(x)\,dx\]?
Under which condition does \[P_6\] give \[\int_{0}^{2a} f(x)\,dx=0\]?
Which expression is equal to \[\frac{1}{1+\sqrt{\tan x}}\] in the evaluation of \[\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\]?
If \[I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\], what is the value of \[I\]?
