Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]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 (x dx)/(1+ sin x)`
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^(2x) cos^5 xdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x`.
Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x` ......(i)
Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`, we get
I = `int_2^5 ("( )")/(sqrt(7 - x) + "( )") "d"x` ......(ii)
Adding equations (i) and (ii), we get
2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x)) "d"x + ( ) "d"x`
2I = `int_2^5 (("( )" + "( )")/("( )" + "( )")) "d"x`
2I = `square`
∴ I = `square`
`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_0^1 (1 - x)^5`dx = ______.
If `int_0^"a" sqrt("a - x"/x) "dx" = "K"/2`, then K = ______.
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^pi x*sin x*cos^4x "d"x` = ______.
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
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.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate:
`int_0^1 |2x + 1|dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
