Advertisements
Advertisements
Question
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Advertisements
Solution
We have I = `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x` ....(1)
= `int_0^(pi/2) (tan^7(pi/2 - x))/(cot^7(pi/2 - x) + tan^7(pi/2 - x)) "d"x` ......By (p4)
= `int_0^(pi/2) (cot^7 (x) "d"x)/(cot^7x "d"x + tan^7x)` .....(2)
Adding (1) and (2), we get
2I = `int_0^(pi/2) ((tan^7x + cot^7x)/(tan^7x + cot^7x))"d"x`
= `int_0^(pi/2) "d"x` which gives I = `pi/4`
APPEARS IN
RELATED QUESTIONS
Evaluate : `intlogx/(1+logx)^2dx`
Evaluate : `intsec^nxtanxdx`
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
`int_2^4 x/(x^2 + 1) "d"x` = ______
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_2^3 x/(x^2 - 1)` dx = ______
`int_-9^9 x^3/(4 - x^2)` dx = ______
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_0^1 log(1/x - 1) "dx"` = ______.
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_0^1 1/(2x + 5) dx` = ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
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`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
Which property is represented by \[\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt\]?
Select the formula for \[P_2\] : Splitting the Interval.
For an Even Function, which condition is satisfied?
