मराठी

Evaluate d∫0π2tan7xcot7x+tan7xdx

Advertisements
Advertisements

प्रश्न

Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`

बेरीज
Advertisements

उत्तर

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`

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 7: Integrals - Solved Examples [पृष्ठ १५१]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12
पाठ 7 Integrals
Solved Examples | Q 10 | पृष्ठ १५१

संबंधित प्रश्‍न

Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`


Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`


By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) cos^2 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^a  sqrtx/(sqrtx + sqrt(a-x))   dx`


`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.


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))` .   


`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?


`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?


`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______


The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.


`int_2^3 x/(x^2 - 1)` dx = ______


`int_0^{pi/2} xsinx dx` = ______


`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?


`int_0^1 log(1/x - 1) "dx"` = ______.


`int_0^pi x sin^2x dx` = ______ 


`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.


Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`


`int_(-2)^2 |x cos pix| "d"x` is equal to ______.


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)


Evaluate:

`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`


`int (dx)/(e^x + e^(-x))` is equal to ______.


The value of `int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))  dx` is


Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`


If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0

⇒ `1/4 (square - square)` = 0

⇒ b4 – `square` = 0

⇒ (b2 – a2)(`square` + `square`) = 0

⇒ b2 – `square` = 0 as a2 + b2 ≠ 0

⇒ b = ± `square`


If `intxf(x)dx = (f(x))/2` then f(x) = ex.


Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?


`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.


If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.


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 ______.


If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.


`int_0^(π/4) x. sec^2 x  dx` = ______.


`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.


Evaluate the following limit :

`lim_("x"->3)[sqrt("x"+6)/"x"]`


Evaluate the following integral:

`int_0^1 x(1-x)^5 dx`


Solve the following.

`int_0^1 e^(x^2) x^3dx`


Evaluate the following integrals:

`int_-9^9 x^3/(4 - x^3 ) dx`


Evaluate:

`int_0^sqrt(2)[x^2]dx`


The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×