you might then have to restore to completion of squares in some cases after the manipulation
in some cases the manipulation can be done through inspection
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usually u is chosen so that successive derivatives of u will become zero after a finite number of steps
provided v has a nice integral or you can make the choice using ILATE rule
inverse , log, algebraic, trigonometric, exponential
for eg. for integral of [x² cosx]
u = x² , v = cosx
but for integral of [x² ln(x)] , you should make it into integral of [ ln(x) * x²]
so that u = ln(x) , v = x²
for integral of lnx, make it into integral of [lnx * 1] with u = lnx , v = 1
in some integrals, like integral of [ (e^x) cosx ], take the integral as I, apply integration by parts a couple of times or so, and then obtain the original integral I again, then rearrange and extract the value of I.
also note the type
here split into two integrals
use integration by parts once on integral of [f(x)exp(x)] taking f(x) as the first function
this will cancel off the integral of [f '(x)exp(x)] leaving you with exp(x) * f(x ) + C which is the required answer Examples on integration by parts
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some interesting things to do with your old computer
1. Converting the old computer into a useful multimedia player :
first of al check if your old computer runs winamp or any other audio playback software. It is usually better to use older versions of winamp which can usually be found at (oldapps.com) at (oldversion.com) . For better quality playback , the soundcard of the computer can be connected to the aux input of stereo players.Using a tv-tuner card, the old computer can also be used for watching tv .
2.Using the computer as a firewall,proxy server etc :
You can change your old computer into a firewall by using software like smoothwall (smoothwall.org). You can also use it to share a single internet connection with more than one computer by using proxy server software like wingate (www.wingate.com) , freeproxy (handcraftedsoftware.org) etc on the old computer. If you are interested in helping people circumvent censorship and have a static ip address you can also use the old computer for running censorship circumventing software ( www.peacefire.org ) on the old computer (but you have to be careful of spammers and other legal problems that may arise).
3. experiment by installing win95/98, linux or dos on the old computer .
The old computer can be used to run old software, games which might not run on new operating systems. If you wanted to experiment by installing linux or other such operating systems, the old computer will be a nice tool for that.
4. changing the old computer into a file server or printer server.
you can use the old computer as a server to store commonly used documents and mp3s. This makes it easier to backup those files. Instead of many printers connected with different computers, you can set up a network so that printing takes place on printers connected to the centralised old computer. This is useful especially if you routinely print hundreds of pages. Some of the older dot matrix printers can be connected only to serial ports.(some older accounting software write only to such dot matrix printers). The old computer (usually will have serial ports) can be used to connect such old printers .
5. Study hardware of computers
Most people will be curious about the internals of a computer, but may not be courageous to open up a new branded pc. The old computer is a nice tool to practice assembling a computer by yourself. Very good free tutorials are available on the internet.
If you are not interested in such things, you could think of donating the old computer to the needy or you could recycle the old computer. Do not throw it out as trash since the computer usually contains a lot of toxic material like lead, antimony, arsenic, selenium, mercury etc. (news.bbc.co.uk) (svtc.etoxics.org) . Be careful and try to permanently erase the contents of the old hard disk using software like eraser (sourceforge.net) , if you are giving away /donating your old computer . An article about the dangers of improper erasure of data is available here.
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arc length of x= t²+1 , y= 2t-3 , 0≤t≤1 ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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let A = arctan(4/3) tanA = 4/3 draw a rough right triangle to get
sinA = 4/5 ; cosA = 3/5
let B = arccos(12/13) as before cosB = 12/13 ; sinB = 5/13
now sin [arctan(4/3)-arccos(12/13)] = sin[A-B] = sinAcosB - cosAsinB trigonometric identities =(4/5 )(12/13) - (3/5)(5/13) = 33 / 65
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free online guide to certain topics on integration for plus two, isc , cbse etc
First of all try to familiarize yourself with the formulae on integration and trigonometric identities Basic Integration you can integrate an expression term by term
that is you can break up an integral of sum or difference of two expressions into sum or difference of two integrals but do not do that with a product or quotient of two functionsbut integral of 2f(x) can be expressed as 2 * integral of f(x) since "2" is a constantDefinite Integrals (Integrals with limits) trying to write expressions in the form x^n
for eg.
cube root of x can be written as x^(1/3)
1 / (x ³) can be written as x^(-3)
if you know that the integral of f(x) is g(x) + C
then the integral of f(ax+b) will be ( 1/a ) g(ax+b) + C
you can divide the numerator term by term using the denominator but not the other way.if the numerator is of higher degree than the denominator use long division if you have a product of sine and/or cosine terms use trigonometric identities for sinAcosB etc to break them into sum or difference before integrating
you can use substitution methods or trigonometric identities to evaluate integrals with powers of sine or cosines.
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visit to trivandrum zoo 5a leopard atop a tree ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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visit to the zoo 4a couple of deer fighting it out ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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visit to trivandrum zoo 3a giraffe with a bandage on its leg ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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visit to trivandrum zoo 2many bats enjoying their siesta ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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a visit to the trivandrum zoo 1some deer enjoying their lunch
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finding a perpendicular vector using the concept of dot product find a vector perpendicular to both [ -1, 3, 4 ] and [-2, -1, 3]
let [x ,y, z] be the required vector
since [x ,y, z] is perpendicular to both [ -1, 3, 4 ] and [-2, -1, 3] -1x + 3y + 4z = 0 -2x -1y + 3z = 0
three unknowns and two equations choose one unknown as arbitrary
put x = a
3y + 4z = a -y +3z = 2a
solving
y = -5a / 13 z = 7a / 13
so if a = 13
x = 13 , y = -5 , z = 7
therefore [13 , -5 , 7] or any of it's scalar multiples will be the required vector ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems disclaimer: There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work
derivative of sqrt(3x) from first principles let f(x) = sqrt(3x) replace x with x+h to get f(x+h) take the difference f(x) - f(x+h) introduce the conjugate before taking limit as h--> 0 ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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If G is an abelian group , let f : G -->G be defined by f(x) = xֿ¹ show that f is an automorphism on G ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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volume using integral calculus Volume of the solid obtained by revolving the shaded region about the x-axis is
solved problems on finding volume of solid of revolution using integration
volume of the solid obtained by revolving the ellipse ( x² / a²) + ( y² / b²) = 1 about the x axis
volume of the solid obtained by revolving the cardioid r = a(1 + cosθ ) about the initial line
volume of the spindle shaped solid obtained by revolving the
hypocycloid x^(2/3) + y^(2/3) =a^(2/3) about the x axis.
calculation of area using integration (calculus method) The area of the region bounded by y = f(x) , the x-axis , the horizontal lines x = a , x = b is given by
show that e^(ix) = cosx + i sinx using power series ----------------------------------------------------------- please leave your comments below ------------------------------------------------------------ index of math problems
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you might then have to restore to completion of squares in some cases after the manipulation
