Advertisements
Advertisements
प्रश्न
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
उत्तर
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = `1/2`.
Explanation:
Given that `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`
⇒ `1/4 int_0^"a" 1/((1/4 + x^2)) "d"x = pi/8`
⇒ `int_0^pi 1/([(1/2)^2 + x^2]) "d"x = pi/2`
⇒ `1/(1/2) [tan^-1 x/(1/2)]_0^"a" = pi/2`
⇒ `2[tan^-1 2"a" - tan^-1 0] = pi/2`
⇒ `tan^-1 2"a" = pi/4`
⇒ 2a = `tan pi/4`
⇒ 2a = 1
⇒ a = `1/2`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `intsec^nxtanxdx`
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) 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`
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
Evaluate`int (1)/(x(3+log x))dx`
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^{pi/2} log(tanx)dx` = ______
`int_0^{pi/2} cos^2x dx` = ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Solve the following.
`int_2^3x/((x+2)(x+3))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 definite integral:
`int_-2^3(1)/(x + 5) dx`
