Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
Advertisements
उत्तर
Let `I = int_0^(pi/2) (2 log sin x - log sin 2x) dx`
`= int_0^(pi/2) [2 log sin x - log (2 sin x cos x)] dx`
`= int_0^(pi/2) [2 log sinx - log 2 - log sin x - log cos x] dx`
`= int_0^(pi/2) [log sin x - log 2 - log cos x] dx`
`= int_0^(pi/2) log sin x dx - int_0^(pi/2) log 2 dx - int_0^(pi/2) log cos x dx`
`= int_0^(pi/2) log sin x dx - int_0^(pi/2) log 2 dx - int_0^(pi/2) log cos (pi/2 - x) dx` `....[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
`= int_0^(pi/2) log sinx dx - (log 2) [x]_0^(pi/2) - int_0^(pi/2) log sin x dx`
`= - (log 2) (pi/2 - 0)`
`= pi/2 log2`
`= pi/2 log (2)^-1`
`= pi/2 log (1/2)`
APPEARS IN
संबंधित प्रश्न
Evaluate : `intsec^nxtanxdx`
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Find `dy/dx, if y = cos^-1 ( sin 5x)`
`int_0^1 "e"^(2x) "d"x` = ______
`int_0^{pi/2} log(tanx)dx` = ______
`int_0^{pi/2} xsinx dx` = ______
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` 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:
`int (dx)/(e^x + e^(-x))` is equal to ______.
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
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 ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
What is the value of a definite integral when its upper and lower limits are equal?
When is \[P_4\] : Special Case of \[P_3\] applied?
