Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
Advertisements
उत्तर
Let `I = int_0^(pi/2) sqrtsinx/(sqrt sinx + sqrt cos x) dx` ...(i)
Replace x to `(pi/2 - x)` in (i)
`[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
`I = int_0^(pi/2) (sqrt sin (pi/2 - x))/ (sqrt sin (pi/2 - x) + sqrt cos (pi/2 - x)) dx`
`I = int_0^(pi/2) sqrtcosx/(sqrtcos x + sqrt sin x) dx` ...(ii)
Adding (i) and (ii), we get
`2I = int_0^(pi/2) [sqrt sinx/ (sqrt sinx + sqrt cos x) + sqrt cos x/(sqrt cos x + sqrt sinx)] dx`
`= int_0^(pi/2) (sqrt cos x + sqrt sin x)/(sqrt cosx + sqrt sin x)`
`= int_0^(pi/2) dx = [x]_0^(pi/2)`
`= pi/2 - 0`
`= pi/2`
⇒ `I = pi/4`
APPEARS IN
संबंधित प्रश्न
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_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| 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.
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total revenue R is increasing.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_2^3 x/(x^2 - 1)` dx = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^{pi/2} xsinx dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_0^pi x*sin x*cos^4x "d"x` = ______.
Which of the following is true?
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate: `int_0^π x/(1 + sinx)dx`.
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate: `int_-1^1 x^17.cos^4x dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
