Advertisements
Advertisements
Question
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
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
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_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
`int_2^4 x/(x^2 + 1) "d"x` = ______
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int (cos x + x sin x)/(x(x + cos x))`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_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
