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
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_0^{pi/2} xsinx dx` = ______
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_0^1 log(1/x - 1) "dx"` = ______.
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
`int_0^9 1/(1 + sqrtx)` dx = ______
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
`int_4^9 1/sqrt(x)dx` = ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
