Advertisements
Advertisements
प्रश्न
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Advertisements
उत्तर
Let I = `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x)` ...(i)
Using property `int_a^b f(x)dx = int_a^b f(a + b - x)dx`, we get
I = `int_1^3 sqrt(4 - x)/(sqrt(4 - x) + sqrt(x))dx` ...(ii)
On adding equations (i) and (ii}, we get
2I = `int_1^3 (sqrt(x) + sqrt(4 - x))/(sqrt(x) + sqrt(4 - x))dx`
= `int_1^3 1dx`
= `[x]_1^3`
= 3 – 1 = 2
∴ I = 1
APPEARS IN
संबंधित प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
`int_"a"^"b" "f"(x) "d"x` = ______
Evaluate `int_0^1 x(1 - x)^5 "d"x`
`int_0^1 (1 - x)^5`dx = ______.
The value of `int_1^3 dx/(x(1 + x^2))` is ______
Which of the following is true?
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
`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:
If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0
⇒ `1/4 (square - square)` = 0
⇒ b4 – `square` = 0
⇒ (b2 – a2)(`square` + `square`) = 0
⇒ b2 – `square` = 0 as a2 + b2 ≠ 0
⇒ b = ± `square`
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.
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 ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
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_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
