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 : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_0^{pi/2} xsinx dx` = ______
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^pi sin^2x.cos^2x dx` = ______
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
Evaluate: `int_(-1)^3 |x^3 - x|dx`
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 ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
Which formula is \[P_5\] : Halving the Upper Limit (Addition)?
For an Even Function, which condition is satisfied?
Why can \[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^{2}x\,dx\] be written as \[2\int_{0}^{\frac{\pi}{4}}\sin^{2}x\,dx\]?
What is the value of \[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^{2}x\,dx\]?
