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_0^1 x(1-x)^n dx`
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`
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
`int_2^4 x/(x^2 + 1) "d"x` = ______
Evaluate `int_0^1 x(1 - x)^5 "d"x`
`int_0^1 (1 - x)^5`dx = ______.
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
Which of the following is true?
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
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_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int_0^1 1/(2x + 5) dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
If f(x) = `(2 - xcosx)/(2 + xcosx)` and g(x) = logex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
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 the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is ______.
