Advertisements
Advertisements
Question
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
Solution
`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
RELATED QUESTIONS
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_"a"^"b" "f"(x) "d"x` = ______
`int_0^1 "e"^(2x) "d"x` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_0^{pi/2} log(tanx)dx` = ______
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_0^{pi/2} cos^2x dx` = ______
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
`int_0^1 1/(2x + 5) dx` = ______.
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
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 ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
