हिंदी

Aafd∫-aaf(x)dx = 0 if f is an ______ function.

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प्रश्न

`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.

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उत्तर

`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an Odd function.

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अध्याय 7: Integrals - Solved Examples [पृष्ठ १६३]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12
अध्याय 7 Integrals
Solved Examples | Q 30 | पृष्ठ १६३

संबंधित प्रश्न

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)  sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx` 


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^(pi/4) log (1+ tan x) dx`


Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`


Evaluate : `int 1/("x" [("log x")^2 + 4])  "dx"`


Evaluate = `int (tan x)/(sec x + tan x)` . dx


Using properties of definite integrals, evaluate 

`int_0^(π/2)  sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`


Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`


`int_2^4 x/(x^2 + 1)  "d"x` = ______


`int_(-7)^7 x^3/(x^2 + 7)  "d"x` = ______


By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x`.

Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x`     ......(i)

Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x)  "d"x`, we get

I = `int_2^5 ("(  )")/(sqrt(7 - x) + "(  )")  "d"x`   ......(ii)

Adding equations (i) and (ii), we get

2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x))  "d"x + (   )  "d"x`

2I = `int_2^5 (("(    )" + "(     )")/("(    )" + "(     )"))  "d"x`

2I = `square`

∴ I =  `square`


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


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


`int_0^1 (1 - x)^5`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 x tan^-1x  dx` = ______ 


`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______ 


`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`


The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______ 


`int_0^1 "e"^(5logx) "d"x` = ______.


`int_(-1)^1 (x + x^3)/(9 - x^2)  "d"x` = ______.


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


`int_(-5)^5  x^7/(x^4 + 10)  dx` = ______.


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


Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.


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 ______.


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


Evaluate the following integral:

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


 `int_-9^9 x^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:

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


Solve the following.

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


`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =


What is the value of a definite integral when its upper and lower limits are equal?


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