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महाराष्ट्र राज्य शिक्षण मंडळएचएससी वाणिज्य (इंग्रजी माध्यम) इयत्ता १२ वी

∫-77x3x2+7 dx = ______ - Mathematics and Statistics

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प्रश्न

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

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पाठ 1.6: Definite Integration - Q.2

संबंधित प्रश्‍न

 
 

Evaluate : `intlogx/(1+logx)^2dx`

 
 

If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`

(A) 1

(B) 2

(C) –1

(D) –2


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

`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) 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_2^8 |x - 5| dx`


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_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.


\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.


Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .


Evaluate : `int _0^(pi/2) "sin"^ 2  "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))` .   


The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total  revenue R is increasing.


Find `dy/dx, if y = cos^-1 ( sin 5x)`


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


Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.


`int_0^2 e^x dx` = ______.


`int_1^2 1/(2x + 3)  dx` = ______


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^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______ 


The value of `int_1^3 dx/(x(1 + x^2))` is ______ 


`int_{pi/6}^{pi/3} sin^2x dx` = ______ 


`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______ 


`int_(pi/4)^(pi/2) sqrt(1-sin 2x)  dx =` ______.


`int_0^9 1/(1 + sqrtx)` dx = ______ 


`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.


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


`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.


`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.


Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.


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 ______.


What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?


Evaluate `int_0^(π//4) log (1 + tanx)dx`.


If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______


`int_1^2 x logx  dx`= ______


`int_0^(2a)f(x)/(f(x)+f(2a-x))  dx` = ______


Solve the following.

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


Evaluate the following integral:

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


Evaluate the following definite integral:

`int_-2^3(1)/(x + 5)  dx`


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