Ok here's one,
Intergrate xlnx

^^^ Hurray! The thread has been resurrected!!

@ inesQ: i dey feel you. I've been trying to understand combinatrics for like forever but you go through it like you invented it. Nice one o!
There is no question for me to answer so i'll go ahead and ask mine. Prove the almighty formula. I cant type the formula cos i'm browsing from a phone!

Chidor6:

Ok here's one,
Intergrate xlnx

We will integrate by parts using the rule
∫vdu = uv - ∫vdu
where v, u are functions of the independent variable under integration

here we choose du as lnxdx and v as x since v is easily differentiated and du is to be integrated (integrating x will lead to more instability in the integral)

du = lnx dx i.e. u = ∫lnx dx
v = x i.e. dv = dx

for u = ∫lnxdx  lets integrate by parts again thus
u = xlnx - x

∫xlnx dx = (xlnx -x)(x) - ∫(xlnx - x)dx

∫xlnx dx = x²lnx - ∫xlnx dx + ∫xdx

Let I = ∫xlnxdx

i.e. I = x²lnx - x² - I + 0.5x² + k

i.e. collecting like terms

2I = x²lnx - x² + 0.5x² + k

I = 0.5 (x²lnx - 0.5x² + k)

Therefore ∫xlnxdx = 0.5x²lnx - 0.25x² + k' where k' = 0.5k

or making it neat

∫xlnxdx = 0.25x² (2lnx - 1) + C

we can check our answers here, it computes online with MATHEMATICA
http://integrals.wolfram.com/index.jsp

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medjai:

@ inesQ: i dey feel you. I've been trying to understand combinatrics for like forever but you go through it like you invented it. Nice one o!
There is no question for me to answer so i'll go ahead and ask mine. Prove the almighty formula. I cant type the formula cos i'm browsing from a phone!

The "almighty" formula is the quadratic formula x=(-b±√(b²-4ac))/2a for the quadratic equation ax² + bx + c =0

Typing all the symbols will be too much stress so I have added it in an image below. Hope you can view it on your mobile device

@Topic: Can't think of any question right now. . .

^^^ kai I just found out that it would have been far simpler to swap the integral parts u and v above. My bad

q.e.d :d

Lol, ur bad, the long form's way better 4 structured questions.

Try dis,
Find the area under the curve for y=1/2(e^2x) with boundaries from 0 to 2 along the x-axis.

@medjai: Did you know that you can use a similar method used for those quadratic equations, in cubic equations?

i.e. ax³+bx²+cx+d=0

So, those that call the quadratic formula the almighty formula, it's because they haven't seen more complex polynomial formulae!  

I have attached Cardan's formula for cubic equations below. . . the proof is a little insane  

A simpler form is this one:
x   =   {q + [q² + (r-p²)³]^1/2}^1/3   +   {q - [q² + (r-p²)³]^1/2}^1/3   +   p

where
p = -b/(3a),   q = p³ + (bc-3ad)/(6a²),   r = c/(3a)

So which one is the almighty formula? The one for fourth power polynomials is even more insane. . . you can check it out here, it may blow your mind http://planetmath.org/encyclopedia/QuarticFormula.html (don't check it on a phone, because even on a PC you will scroll and scroll )

You're Einstein at large. Maybe the 'Almighty formula' should be renamed child's play formula. Did you major in maths?

^^^Ha abeg o. I'm just a graduate engineer with a passion for mathematics