Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
Evaluate:
`int_0^(pi/4) log (1+ tan x) dx`
Advertisements
उत्तर
Let I = `int_0^(pi/4) log (1 + tan x) dx` ....(1)
∴ I = `int_0^(pi/4) log [1 + tan (pi/4 - x)] dx` `...[int_0^a f(x) dx = int_0^a f(a - x) dx]`
⇒ I = `int_0^(pi/4) log {1 + (tan pi/4 - tan x)/(1 + tan pi/4 tan x)}dx`
⇒ I = `int_0^(pi/4) log {1 + (1 - tan x)/(1 + tan x)} dx`
⇒ I = `int_0^(pi/4) log 2/((1 + tan x)) dx`
⇒ I = `int_0^(pi/4) log 2 dx - int_0^(pi/4) log (1 + tan x) dx`
⇒ I = `int_0^(pi/4) log 2 dx - I` ...[From (1)]
⇒ 2I = `[x log 2]_0^(pi/4)`
⇒ 2I = `pi/4 log 2`
⇒ I = `pi/8 log 2`
APPEARS IN
संबंधित प्रश्न
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate : ∫ log (1 + x2) dx
`int_0^1 "e"^(2x) "d"x` = ______
`int_1^2 1/(2x + 3) dx` = ______
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_0^1 log(1/x - 1) "dx"` = ______.
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^1 "e"^(5logx) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
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.
`int_0^1|3x - 1|dx` equals ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
