Advertisements
Advertisements
प्रश्न
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Advertisements
उत्तर
We have I = `int_0^(pi/2) (sin^2x)/(sinx + cosx) "d"x`
= `int_0^(pi/2) (sin^2(pi/2 - x))/(sin(pi/2 - x) + cos(pi/2 - x)) "d"x` ....(By P4)
⇒ I = `int_0^(pi/2) (cos^2x)/(sinx + cosx) "d"x`
Thus, we get 2I = `1/sqrt(2) int_0^(pi/2) ("d"x)/(cos(x - pi/4))`
= `1/sqrt(2) int_0^(pi/2) sec(x - pi/2) "d"x`
= `1/sqrt(2) [log(sec(x - pi/4) + tan(x - pi/4))]_0^(pi/2)`
= `1/sqrt(2)[log(sec pi/4 + tan pi/4) - log sec(- pi/4) + tan(- pi/4)]`
= `1/sqrt(2) [log(sqrt(2) + 1) - log(sqrt(2) - 1)]`
= `1/sqrt(2) log|(sqrt(2) + 1)/(sqrt(2) - 1)|`
= `1/sqrt(2) log((sqrt(2) - 1)^2/1)`
= `2/sqrt(2) log(sqrt(2) + 1)`
Hence I = `1/sqrt(2) log(sqrt(2) + 1)`.
APPEARS IN
संबंधित प्रश्न
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
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`
`int_0^2 e^x dx` = ______.
`int_1^2 1/(2x + 3) dx` = ______
`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^1 (1 - x)^5`dx = ______.
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" 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 ______.
`int (dx)/(e^x + e^(-x))` is equal to ______.
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
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_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
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 ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate `int_1^2(x+3)/(x(x+2)) 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`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
