Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\int \sqrt{x - x^2} \text{ dx }\]
\[ = \int \sqrt{- \left( x^2 - x \right)} \text{ dx }\]
\[ = \int \sqrt{- \left\{ x^2 - x + \left( \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2 \right\}} \text{ dx }\]
\[ = \int \sqrt{\left( \frac{1}{2} \right)^2 - \left( x - \frac{1}{2} \right)^2} dx\]
\[ = \left( \frac{x - \frac{1}{2}}{2} \right) \sqrt{x - x^2} + \frac{1}{8} \sin^{- 1} \left( \frac{x - \frac{1}{2}}{\frac{1}{2}} \right) + C \left[ \because \int\sqrt{a^2 - x^2}dx = \frac{1}{2}x\sqrt{a^2 - x^2} + \frac{1}{2} a^2 \sin^{- 1} \frac{x}{a} + C \right]\]
\[ = \left( \frac{2x - 1}{4} \right) \sqrt{x - x^2} + \frac{1}{8} \text{ sin}^{- 1} \left( 2x - 1 \right) + C\]
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`1/(1 - tan x)`
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate `int (5"x" + 1)^(4/9)` dx
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int cos^7 x "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (cos x)/(1 - sin x) "dx" =` ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int (logx)^2/x dx` = ______.
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
