Advertisements
Advertisements
Question
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Options
2
`3/4`
0
- 2
Advertisements
Solution
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is 0.
Explanation:
Let I `= int_0^(pi//2) log ((4 + 3 sin x)/(4 + 3 cos x)) "dx"`
Also, `I = int_0^(pi/2) log [(4+3 sin (pi/2 - x))/(4 + 3 cos (pi/2 - x))] dx`
`[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
⇒ ` I = int_0^(pi/2) log [(4+3 cos x)/(4+3 sin x)] dx`
⇒ `I = - int_0^(pi/2) log [(4+3sinx)/(4+3cosx)] dx`
⇒ I = -I
⇒ 2I = 0
⇒ I = 0
APPEARS IN
RELATED QUESTIONS
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_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total revenue R is increasing.
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
`int_0^1 "e"^(2x) "d"x` = ______
`int_1^2 1/(2x + 3) dx` = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^{pi/2} cos^2x dx` = ______
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)
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int (dx)/(e^x + e^(-x))` is equal to ______.
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_0^1 1/(2x + 5) dx` = ______.
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` 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_0^(π/4) x. sec^2 x dx` = ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is ______.
