Advertisements
Advertisements
Question
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Advertisements
Solution
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
RELATED QUESTIONS
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
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_((-pi)/2)^(pi/2) sin^2 x dx`
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.
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
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 = ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_0^(π/4) x. sec^2 x dx` = ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
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 the following integral:
`int_0^1 x(1 - 5)^5`dx
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
`int_-9^9 x^3/(4-x^2) 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:
`int_0^sqrt(2)[x^2]dx`
