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
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
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_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Evaluate`int (1)/(x(3+log x))dx`
`int_"a"^"b" "f"(x) "d"x` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_0^pi x*sin x*cos^4x "d"x` = ______.
Which of the following is true?
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`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:
Evaluate: `int_(-1)^3 |x^3 - x|dx`
`int_4^9 1/sqrt(x)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 ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following definite integral:
`int_1^3 log x dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
