Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Advertisements
उत्तर
Let I = `int_0^a (sqrtx)/(sqrtx + sqrt(a - x)) dx` ....(i)
`= I = int_0^a (sqrt(a - x))/(sqrt(a - x) + sqrt (a - (a - x)))`
I = `int_0^a sqrt(a - x)/(sqrt(a - x) + sqrtx) dx` ....(ii)
`[because int_0^a f(x) dx = int_0^a f(a - x) dx]`
On adding equation (i) and (ii),
2 I = `int_0^a (sqrtx + sqrt(a - x))/(sqrt(a - x) + sqrtx) dx`
2 I `= int_0^a 1 * dx => [x]_0^a`
⇒ 2I = a
∴ `I = a/2`
APPEARS IN
संबंधित प्रश्न
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_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| 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 "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate : ∫ log (1 + x2) dx
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_"a"^"b" "f"(x) "d"x` = ______
`int_0^1 "e"^(2x) "d"x` = ______
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_0^{pi/2} cos^2x dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
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)`
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Evaluate: `int_(-1)^3 |x^3 - x|dx`
`int_0^1 1/(2x + 5) dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
`int_1^2 x logx dx`= ______
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Which formula expresses \[P_1\] : Reversing Limits?
When is \[P_4\] : Special Case of \[P_3\] applied?
