Advertisements
Advertisements
प्रश्न
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
विकल्प
2
`3/4`
0
- 2
उत्तर
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
संबंधित प्रश्न
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_1^2 1/(2x + 3) dx` = ______
`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_0^{pi/2} log(tanx)dx` = ______
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^{pi/2} cos^2x dx` = ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`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_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
Evaluate: `int_(-1)^3 |x^3 - x|dx`
`int_4^9 1/sqrt(x)dx` = ______.
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
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 ______.
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 ______.
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 `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
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`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
