How do you integrate 1/(x*(x-a))?
105440
You can try and find constants A and B such that\n\n1/(x(x-a)) = A/x + B/(x-a)\n\nIf you get this, then the integration is simple.\n\nHow do you get it?\n\nStart from A/x + B/(x-a) and calculate the common denominator:\n[A(x-a) + Bx]/[x(x-a)] = [(A+B)x - aA]/(x(x-a))\n\nNow to get the expression equal to 1/(x(x-a)) it must be:\n\nA+B = 0  ---> A= -B\naA = 1  ---> A = 1/a\n\nso \nA = 1/a (constant)\nB= - 1/a\n\nsubstitute and you get the integral as the sum of two simpler ones which get two logarithms by integration.\n\nThen you apply the property of the sum of logarithms to obtain a single one.\n\nI hope it's clear.
842823
You start by finding an antiderivative of each one (not difficult, just break up the fraction and use the power rule on each part). Then you evaluate it at the limits of integration, and subtract. For example:\n\n[1, 8]∫(6x²+2)/√x dx\n[1, 8]∫6x^(3/2) + 2x^(-1/2) dx\n12/5 x^(5/2) + 4x^(1/2) from 1 to 8:\n12/5 (8)^(5/2) + 4(8)^(1/2) - (12/5 (1)^(5/2) + 4(1)^(1/2))\n1536√2/5 + 8√2 - (12/5 + 4)\n1576√2/5 + 32/5\n\nYou can do the second one on your own.
217977
f(E) = 1/(exp{(E-Ef)/kT}+1)
957759
a. 0 = 7-x^2, x^2=7, x = +/-sqrt(7)\nArea = Integral[-sqrt(7), sqrt(7)] (7-x^2)dx = 7x - (1/3)x^3 |-sqrt(7), sqrt(7) = 2[7sqrt(7) - (1/3)sqrt(7)^3] ~= 24.69 u^2\n\nb. 0 = x^4-8, x^4 = 8, x = +/-8^(1/4)\nArea = (1/5)x^5-8x |-8^(1/4), 8^(1/4) = 2[(1/5)8^(5/4)-8(8)^(1/4)] ~absv= 21.53 u^2
615045
[x/(x+2)]\n= [1/(1+(2/x))] --- divide the numerator & denominator by x.\n\nAs x approaches +ve infinity, 2/x approaches 0.\nSo [x/(x+2)] approaches 1.
117158
the integration >>> increase the power by one and divide by the new power (don't forget the the constant)\n\nx^n -------> x^(n+1)/(n+1)\n\ny = 1/(x^2)\ny = x^-2\nthe integration = x^(-1)/(-1) + c\n= -x^-1 + c\n= -1/x + constant
1079776
you can simply the first one to (x-a)(x-a) then you can divide that by (x-a) and it equals (x-a)\n\nDo you know synthetic division; cuz you can divide the second problem by (x+2)^2 if you mulitply (x+2)^2 out.
700959
(1/x) / (x-6)\n1 / (x^2 - 6x)\n\n(a/b) / (c/d) = \n\n(ad) / (bc)
40514
When you are doing integrations, all you are doing is computing an area under a curve. Sometimes the formula for the area will be obvious to you, from something you already know or recognize. Othertimes, you have to transform the problem into one you know already. \n\nThis is the method called integration by substitution. \n\nFor example, suppose you want to integrate the function sqrt(1 - x^2) with respect to x, between some limits. Most times, you will not remember or recognize this integral. However, you might remember that sqrt(1-(sin a)^2) is cos a, which you know how to integrate, and therefore if you let x=sin a, you have transformed the integral into one you recognize and can easily solve. \n\nThere is a HUGE amount of material written on this on the internet and in books, but this should get you started. All you are doing is transforming the problem into one that you can solve easier.
84808
multiply and divide by (1 - sinx)\n\nyou get (1 - sin^2(x))/(cosx(1-sinx)) that is cos^2(x)/(cosx(1-sinx)) that is cosx/(1-sinx)\n\nnow at the denominator you have a function and at the numerator you have the derivative of that function, save for the sign. \nyour integral is \n-ln(|1-sinx|) + c\n\n---- \n\nproof that we're all saying the same thing:\n- ln |1-sinx| = ln (|1 - sinx|^-1) =\n\n= ln |1/(1-sinx)| ---> divide and multiply again for \n(1 + sinx)  ----> = ln |(1 + sinx)/cos^2(x)|\n\n= ln |(secx + tanx)/cosx| = ln|(secx + tanx)secx| = \n= ln |secx + tanx| + ln|secx|\n\nSo you have many ways to write your answer: you can choose the one which you prefer\n- by elegance\n- by minimization of spread error in calculating it\n- by whatever you like\n\nhave a nice day
105048
Directly, there is no solution.  If you use a series expansion around x=0, \nsin(x)=sum((-1)^n(x^(2n+1))/(factorial(2n+1)))\nDivide this series by x, then integrate.  After the 3rd term, the error term goes to the ten millionths digit, so I'd say you're pretty safe :)\n\nsin(x)/x = 1 - (x^2)/6 + (x^4)/120 - (x^6)/5040 + .....\nIntegrated is:  x - (x^3)/18 + (x^5)/600 - (x^7)/32580 +\nor \nsum((-1)^n)(x^(2n+1))/(2n+1)*factorial(2n+1))
713982
use logarthmic differentiation\n\ny = x^(1/x)\n\nln(y) = (1/x)* ln(x)\n\ndiff w.r.t x\n\n(1/y) (dy/dx) = (1/x)*(1/x) +ln(x)*(-1/x^2)\n\n(dy/dx) = { y/(x^2) }  [ 1 - ln(x) ]\n\nsubstitute for y
664470
Since 1/(a+b+x)=1/a+1/b+1/x\nSo 1/(a+b+x) - 1/x =1/a+1/b\nSo (x-a-b-x)/(x(a+b+x)) = (a+b)/ab\nSo -(a+b)/(a+x+b)=(a+b)/ab\nSo -1/(x(a+b+x))= 1/ab\nSo x(a+b+x) = - ab\nSo x^2+ (a+b)*x + ab =0\nAnalyzing :\n(x+a)(x+b)=0\nEither x = -a\nOr     x + -b
546126
intrgrate o e^(3x)=(1/3)e^(3x)
1006943
Rewrite as a power series, since there is no way to do this using traditional methods:\n\nx^2 - (1/2)x^4 + (3/8)x^6 - (37/144)x^8 + (1031/5760)x^10 -.....\n\nThen integrate:\n\n(1/3)x^3 - (1/10)x^5 + (3/56)x^7 - (37/1296)x^9 + (1031/63360)x^11 -.......
759209
This is a non-elementary integral in x. However, from the way you phrased the problem, I would guess that you want this to compute a double integral: first with repspect to x, then with respect to y. The best way of doing *that* would be to switch the order of integration. Then the integral with respect to y will be xe^(y/x). Be sure to change the limits of integration so that you are integrating over the same region in the xy-plane!
798293
use integration by parts:\n\nu=x\ndv= cos(x) dx\n\ndu =dx\nv= sin(x)\n\nintegral of x cosx dx = xsinx - integral of sin(x) dx\n= x sinx + cos(x) + C\n[