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
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_-9^9 x^3/(4 - x^2)` dx = ______
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_0^pi x*sin x*cos^4x "d"x` = ______.
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
The value of `int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2)) dx` is
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
`int_a^b f(x)dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
Evaluate `int_-1^1 |x^4 - x|dx`.
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Solve the following.
`int_0^1e^(x^2)x^3 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`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
