मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

∫1cosx-sinx dx = _______________ - Mathematics and Statistics

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प्रश्न

`int 1/(cos x - sin x)` dx = _______________

पर्याय

  • `1/sqrt2 log ["cosec" (x + π/4) - cot (x + π/4)] + "c"`

  • `sqrt2 log ["cosec" (x + π/4) + cot (x + π/4)] + "c"`

  • `1/sqrt2 log [sec (x + π/4) + tan (x + π/4)] + "c"`

  • `sqrt2 log [sec (x + π/4) - tan (x + π/4)] + "c"`

MCQ
रिकाम्या जागा भरा

उत्तर

`1/sqrt2 log [sec (x + π/4) + tan (x + π/4)] + "c"`

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पाठ 2.3: Indefinite Integration - MCQ

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संबंधित प्रश्‍न

Show that:  `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`


Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`


Find : `int(x+3)sqrt(3-4x-x^2dx)`


 
 

Evaluate :

`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`

 
 

Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 


Integrate the functions:

`e^(tan^(-1)x)/(1+x^2)`


Integrate the functions:

`(sin^(-1) x)/(sqrt(1-x^2))`


Integrate the functions:

`1/(1 + cot x)`


Integrate the functions:

`sqrt(tanx)/(sinxcos x)`


Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`


Write a value of

\[\int e^x \left( \sin x + \cos x \right) \text{ dx}\]

 


Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]


Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]


\[\text{ If } \int\left( \frac{x - 1}{x^2} \right) e^x dx = f\left( x \right) e^x + C, \text{ then  write  the value of  f}\left( x \right) .\]

Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 


Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`


Evaluate the following integrals:

`int (sin4x)/(cos2x).dx`


Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`


Integrate the following function w.r.t. x:

x9.sec2(x10)


Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1 


Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`


Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`


Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`


Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.


Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`


Integrate the following functions w.r.t.x:

cos8xcotx


Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`


Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`


Evaluate the following : `int  (1)/(x^2 + 8x + 12).dx`


Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`


Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`


Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`


Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`


Evaluate the following integrals :  `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`


Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`


Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`


Evaluate `int 1/("x" ("x" - 1))` dx


If f'(x) = 4x3 − 3x2  + 2x + k, f(0) = 1 and f(1) = 4, find f(x).


Evaluate the following.

`int 1/(sqrt"x" + "x")` dx


Evaluate the following.

`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx


`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c


`int x/(x + 2)  "d"x`


`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1))  "d"x`


State whether the following statement is True or False:

`int3^(2x + 3)  "d"x = (3^(2x + 3))/2 + "c"`


If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.


`int(5x + 2)/(3x - 4) dx` = ______


`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`


`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.


General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)


`int ("d"x)/(x(x^4 + 1))` = ______.


`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.


`int (f^'(x))/(f(x))dx` = ______ + c.


`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.


If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.


`int sqrt(x^2 - a^2)/x dx` = ______.


If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.


`int x/sqrt(1 - 2x^4) dx` = ______.

(where c is a constant of integration)


`int cos^3x  dx` = ______.


Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.


Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.


If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)


Evaluate `int 1/("x"("x" - 1)) "dx"`


Evaluate `int(1 + x + x^2/(2!))dx`


Evaluate the following

`int x^3/sqrt(1+x^4) dx`


If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)


`int (cos4x)/(sin2x + cos2x)dx` = ______.


Evaluate:

`int sin^3x cos^3x  dx`


Evaluate `int(5x^2-6x+3)/(2x-3) dx`


Evaluate the following.

`int 1/ (x^2 + 4x - 5) dx`


Evaluate the following.

`int1/(x^2 + 4x-5)dx`


If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).


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