मराठी

∫ √ X − X 2 D X - Mathematics

Advertisements
Advertisements

प्रश्न

\[\int\sqrt{x - x^2} dx\]
बेरीज
Advertisements

उत्तर

\[\int \sqrt{x - x^2} \text{ dx }\]
\[ = \int \sqrt{- \left( x^2 - x \right)} \text{ dx }\]
\[ = \int \sqrt{- \left\{ x^2 - x + \left( \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2 \right\}} \text{ dx }\]
\[ = \int \sqrt{\left( \frac{1}{2} \right)^2 - \left( x - \frac{1}{2} \right)^2} dx\]
\[ = \left( \frac{x - \frac{1}{2}}{2} \right) \sqrt{x - x^2} + \frac{1}{8} \sin^{- 1} \left( \frac{x - \frac{1}{2}}{\frac{1}{2}} \right) + C \left[ \because \int\sqrt{a^2 - x^2}dx = \frac{1}{2}x\sqrt{a^2 - x^2} + \frac{1}{2} a^2 \sin^{- 1} \frac{x}{a} + C \right]\]
\[ = \left( \frac{2x - 1}{4} \right) \sqrt{x - x^2} + \frac{1}{8} \text{ sin}^{- 1} \left( 2x - 1 \right) + C\]

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 19: Indefinite Integrals - Exercise 19.28 [पृष्ठ १५४]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12
पाठ 19 Indefinite Integrals
Exercise 19.28 | Q 3 | पृष्ठ १५४

संबंधित प्रश्‍न

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`


 
 

Evaluate :

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

 
 

Integrate the functions:

`(x^3 - 1)^(1/3) x^5`


Solve:

dy/dx = cos(x + y)


\[\int\sqrt{3 + 2x - x^2} \text{ dx}\]

Write a value of

\[\int \tan^3 x \sec^2 x \text{ dx }\].

 


Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]


Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]


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

 

 


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


Integrate the following functions w.r.t. x : `cosx/sin(x - a)`


Integrate the following functions w.r.t. x : cos7x


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


Choose the correct options from the given alternatives : 

`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =


Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx


Fill in the Blank.

`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______


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


Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx


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


`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________


Evaluate `int(3x^2 - 5)^2  "d"x`


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


`int (cos x)/(1 - sin x) "dx" =` ______.


`int(log(logx) + 1/(logx)^2)dx` = ______.


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


Evaluate the following.

`int 1/(x^2+4x-5)  dx`


Evaluate `int1/(x(x - 1))dx`


Evaluate the following.

`int x sqrt(1 + x^2)  dx`


Evaluate the following

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


Evaluate:

`int sqrt((a - x)/x) dx`


`int "cosec"^4x  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 (x^3)/(sqrt(1 + x^4)) 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).


Evaluate the following.

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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×