Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\int \sqrt{1 + x - 2 x^2} \text{ dx}\]
\[ = \int \sqrt{2\left( \frac{1}{2} + \frac{x}{2} - x^2 \right)} \text{ dx}\]
\[ = \sqrt{2} \int\sqrt{\frac{1}{2} - \left( x^2 - \frac{x}{2} \right)} \text{ dx}\]
\[ = \sqrt{2} \int \sqrt{\frac{1}{2} - \left( x^2 - \frac{x}{2} + \frac{1}{4^2} - \frac{1}{4^2} \right)} \text{ dx}\]
\[ = \sqrt{2} \int \sqrt{\frac{1}{2} + \frac{1}{16} - \left( x - \frac{1}{4} \right)^2} \text{ dx}\]
\[ = \sqrt{2} \int \sqrt{\left( \frac{3}{4} \right)^2 - \left( x - \frac{1}{4} \right)^2} \text{ dx}\]
\[ = \sqrt{2} \left[ \left( \frac{x - \frac{1}{4}}{2} \right) \sqrt{\left( \frac{3}{4} \right)^2 - \left( x - \frac{1}{4} \right)^2} + \frac{9}{32} \sin^{- 1} \left( \frac{x - \frac{1}{4}}{\frac{3}{4}} \right) \right] + C \left[ \because \int\sqrt{a^2 - x^2}\text{ dx} = \frac{1}{2}x\sqrt{a^2 - x^2} + \frac{1}{2} a^2 \sin^{- 1} \frac{x}{a} + C \right]\]
\[ = \left( \frac{4x - 1}{8} \right) \sqrt{1 + x - 2 x^2} + \frac{9\sqrt{2}}{32} \sin^{- 1} \left( \frac{4x - 1}{3} \right) + C\]
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`(1+ log x)^2/x`
Solve:
dy/dx = cos(x + y)
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
`int logx/(log ex)^2*dx` = ______.
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
`int 1/(xsin^2(logx)) "d"x`
`int sin^-1 x`dx = ?
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate:
`int sin^2(x/2)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate `int1/(x(x-1))dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
For \[t=\cos x\], what is \[dt\]?
What is the value of \[\int\sin^3x\cos^2x\,dx\]?
