Table of Useful Integrals, etc.
xe-
2 a
2a
XP ;xe_3xJ 2a2
4a a
-1 3 5(2n-)1 r 2a+1an
n! 2an+1
x"e~ax dx =_n! an+1 Integration by Parts:
fUdV = UV (VdUU and V are functions of x. Integrate from x = a to x = b
Ssin(ax) dx = - cos(a) a
2
4a
8
16a
4a
where a2 b2
2(a+b
2(a-1
2(a+1)
2(a-) 2(a+1) (xsin(ax) dx=_xsin(2a) cos(2a) 4 4a 8a2
cos(2ax
=x 1 4a 8a3
sih 2a
4a
cos (a+bx xsin (a-xxsin (a+ x 2a+) 2(a-b 2(a+b
xsin(ax sin(b} dx
2(a-
sin(ax) xcos(a) sin(ax dx=
xsin(ax.cos(ax
sin(ax) os(ax) dx
(ax)dx=+sin(2ay) 2 4a
4a
Taylor Series:
Geometric Series:
Euler's Formula: e =coso+isin Quadratic Equation and other higher order polynomials: ax2+bx+c=0 -bb2 - 4ac X 2a
ax4+bx2+c=0 -b vb2 -4ac x=+ 2a
General Solution for a Second Order Homogeneous Differential Equation with Constant Coefficients:
If: y'+ py+qy=0
Assume a solution for y:
y=esxy=sesxy=s2esx s2esx + psesx +qesx =0 O=b+sd+zs pue Hencey=c,e+c,e2
Conversions from spherical polar coordinates into Cartesian coordinates:
x =rsin0cos y =rsinsin x=rcos0 dv = r2 sin0 drdO do oo>I>0 I>0>0 1Z>