Advertisements
Advertisements
Question
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
Advertisements
Solution
Let `"I" = int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"` ...(i)
⇒ `"I" = int_0^pi ((pi-"x")sin(pi-"x"))/(1+3cos^2(pi-"x"))d"x"`
= `int_0^pi (pisin"x")/(1+3cos^2"x")d"x" - int_0^pi (xsin"x")/(1+3cos^2"x")d"x"` ...(ii)
Adding (i) & (ii), we have
we get: `2"I" = int_0^pi(pisin"x")/(1+3 cos^2 "x")` dx
Put cos x = t
⇒ - sin x dx = dt, when x = 0
⇒ t = 1, for x = π ⇒ t = - 1
So, `2I = π int_1^-1 dt/(1 + 3t^2)`
⇒ `π/3 int_-1^1 (dt)/((1/sqrt3)^2 + (t)^2)`
⇒ `π/3 xx sqrt3 [tan^-1(sqrt3t)]_-1^1`
⇒ `(sqrt3π)/3 [ tan^-1sqrt3 - ( - tan^-1 sqrt3)]`
I = `(sqrt3π)/3. π/3 = sqrt3π^2/9`
APPEARS IN
RELATED QUESTIONS
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) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
`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`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
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 ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Evaluate: `int_-1^1 x^17.cos^4x dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
Which property is represented by \[\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt\]?
