Advertisements
Advertisements
प्रश्न
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Advertisements
उत्तर
Let I = `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
= `int_(pi/6)^(pi/3) sqrt(cosx)/(sqrt(sinx) + sqrt(cos x)) dx` ......(i)
Using `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
I = `int_(pi/6)^(pi/3) sqrt(cos(pi/6 + pi/3 - x))/(sqrt(sin(pi/6 + pi/3 - x)) + sqrt(cos(pi/6 + pi/3 - x)))`
I = `int_(pi/6)^(pi/3) sqrt(sinx)/(sqrt(cosx) + sqrt(sinx)) dx` ......(ii)
Adding (i) and (ii), we get
2I = `int_(pi/6)^(pi/3) sqrt(cosx)/(sqrt(sinx) + sqrt(cosx)) dx + int_(pi/6)^(pi/3) sqrt(sinx)/(sqrt(cosx) + sqrt(sinx)) dx`
2I = `int_(pi/6)^(pi/3) dx`
= `[x]_(pi/6)^(pi/3)`
= `pi/3 - pi/6`
= `pi/6`
Hence, I = `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)) = pi/12`
APPEARS IN
संबंधित प्रश्न
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^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_0^1 x tan^-1x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int (dx)/(e^x + e^(-x))` is equal to ______.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
`int_0^1 1/(2x + 5) dx` = ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
`int_0^1|3x - 1|dx` equals ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
