Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
Advertisements
उत्तर
`I = int_0^(pi/2) (sin x - cosx)/(1+sinx cos x) dx` ....(i)
`I = int_0^(pi/2) (sin (pi/2-x)-cos(pi/2-x))/(1 + sin(pi/2-x)cos(pi/2-x))dx`
`I = int_0^(pi/2) (cosx-sinx)/(1+cosxsinx)dx` .....(ii)
Adding (i) and (ii), we get :
`2 I = int_0^(pi/2) ((sin x - cos x)/ (1 + sin x cos x) + (cos x - sin x)/ (1 + sin x cos x)) dx`
`2I = int_0^(pi/2)(sinx-cosx+ cosx - sinx)/(1 +sinxcosx) dx`
`2I = 0 ⇒I=0`
`⇒ int_0^(pi/2) (sinx-cosx)/(1+sinxcosx) dx=0`
APPEARS IN
संबंधित प्रश्न
Evaluate : `intlogx/(1+logx)^2dx`
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_(pi/2)^(pi/2) sin^7 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) 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: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Evaluate`int (1)/(x(3+log x))dx`
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 : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_(-1)^1 log ((2 - x)/(2 + x)) "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)` = ______.
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
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 ?
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate the following definite integral:
`int_4^9 1/sqrt"x" "dx"`
`int_1^2 x logx dx`= ______
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^6 |x + 3|dx`
