Advertisements
Advertisements
प्रश्न
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
उत्तर
`I = int_0^(pi/2) cos^2 x/(1 + sinxcosx)dx` ....(1)
Using `int_0^a f(x) dx = int_0^a f(a -x) dx`
`I = int_0^(pi/2) (cos^2 (pi/2 - x))/(1 + sin(pi/2 -x)cos(pi/2 -x)) dx`
`= int_0^(pi/2) (sin^2 x)/(1+cos x.sin x) dx` .....(2)
Adding eq. (1) & (2)
`2I = int_0^(pi/2) (cos^2x + sin^2 x)/(1+sin xcos x) dx`
`= int_0^(pi/2) 1/(1+sinxcos x) dx``
`= int_0^(pi/2) (sec^2x) /(sec^2x + tan x) dx`
`2I = int_0^(pi/2) (sec^2 x dx)/(1+tan^2 x + tan x)`
Put `tan x = t, sec^2xdx = dt`
when x = 0, t = 0
when `s = pi/2, t = oo`
`2I = int_0^(oo) (dt)/(t^2 + 2t. 1/2+1/4 1/4 + 1)`
`= int_0^(oo) (dt)/((t+1/2)^2 + ((sqrt3)/2)^2`
`= 1/(sqrt3/2) [tan^(-1) ((t+1/2)/(sqrt3/2))]_0^oo`
`= 2/sqrt3 tan^(-1) [(2t + 1)/sqrt3]_0^oo`
`2I = 2/sqrt3 [pi/2 - pi/6]`
`I = 1/sqrt3[(3pi - pi)/6]`
` = 1/sqrt3 [(2pi)/6] = pi/(3sqrt3)`
APPEARS IN
संबंधित प्रश्न
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
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` = ______.
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/2} cos^2x dx` = ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_a^b f(x)dx` = ______.
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_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
Evaluate:
`int_0^1 |2x + 1|dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
