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## George C.'s Resources

Stoichiometry: C3H8  +  5O2  -->  3 CO2  +  4 H2O PV  =  nRT ,  V = nRT/P, (38.8)*10^3 g  C3H8  =  879.9 mole C3H8 reacts with                                 5*(879...

f(t) = e^-(t-x)  sin x dx, Using Euler’s identity, e^im = cos m + i sin m, integrate using the exponential, and take the imaginary part (Im) after integration. f(t) = e^-(t-x) * (e^ix) dx = e^(-t + x + ix) = e^-t * (e^(1 + i)x) dx int f(t) = int [0, t] e^-t * (e^(1...

e^(-10s)/(s(s^2+3s+2)) Factor out e^(-10s) for partial fraction decomposition. 1/(s(s^2+3s+2))  =   1/ s(s+1)(s+2)  =   A/s  +  B/(s+1)  +  C/(s+2) Using the Heaviside coverup method. when s...

Another method, that involves less writing is the Oliver Heaviside coverup method.  Google it and it's well explained.  Just another instrument in your armamentarium for solving partial fraction decomposition.

dx/dy  +    2x  -   (1/y)e^(-2y)  =  0 dx/dy  +   2x   =   (1/y)e^(-2y) x'      +     2x   =   (1/y)e^(-2y) I.F. = e^2y x ...

The absolute value, or modulus of the number z = a + bi is defined by |z| = √(a² + b²) |z1 + z2|  = |(a1 + a2) + (b1 + b2)| = √[(a1 + a2)² + (b1 + b2)²] |z1 + z2|² = (a1 + a2)² + (b1 + b2)²

y'+(1/2)y=2cos(t) vy' + 1/2 vy  =  2 v cos t                 Let  1/2  v  = dv/dt = v' (vy)'  =   2 v cos t                                      ...

You have a typo here.

x-z=arctan(yz) 1 - (dz/dx) =  (y(dz/dx)/(1 + (yz)^2) 1  =  (dz/dx)((1 +  y/(1 + (yz)^2))) dz/dx  =  (1 + (yz)^2)/(1 + y + (yz)^2)

You must paramaterize.  It is easiest if you draw a picture.   [(1/3)^(1/2), 1 ] int (3x - 1/x)dx  +  [1, (3)^(1/2)] int (3/x  -  x)dx   (3/2 x^2 - ln x) [(1/3)^(1/2), 1 ]  +  (3 ln x  -  (x^2/2) [1, (3)^(1/2)]   Performing...

sin x = (e^ix – e^-ix)/2i Let sin x = -4 + sqrt2/2 = k = (e^ix – e^-ix)/2i 2ik = (e^ix – e^-ix) (e^ix – e^-ix) –2ik = 0 e^2ix – 2ike^ix –1 = 0 Let u = e^ix u^2 – 2iku – 1 = 0, u = (2ik ± (((-2ik)^2 – 4(1)(-1))^(1/2))/2 (-2.586i ± 1.64)/2 u=  e^ix  ...

f = ye^x(x)+cos(x).             Subscripts are  tedious.  Let f(xn) be the nth derivative wrt  x. f(x1) = y(xe^x + e^x) - sin x f(x2) = y(xe^x + 2e^x) - cos x f(x3) = y(xe^x + 3e^x) +...

3 vectors defining the plane are ,<2, 0, 0>, <0, 3, 0>, <0, 0, 1> x1 - x2= <2, -3, 0> x1 - x3= <2, 0, -1>  i         j          k 2       ...

dA = 2 pi r dh, the limits of integration in terms of h [0, 22.8]      =  2 pi (12.5 - (12.5/22.8)h) dh  A   = 2 pi (12.5h  - (1/2)(12.5/22.8)h^2)   [0, 22.8]       = 2pi (12.5)(22.8)(1...

This will factor: (sin x - 3)(sin x + 1),  sin x = -1, x = 3pi/2 The solution sin x = 3 has no real solution.

F=<zx^2, z(e^xy^2)-x, x*ln (y^2)> From the paramaterization,  F(r(t)),  <  0,  -cos t,  (cos t) ln (sin t)^2>, (everything multiplied by z vanishes) r'(t) = i  (- sin t) dt   +  j  (cos t) dt + k (0) Int [...

(h - k)/a  =  b^(cx + d) ln ((h - k)/a) =  (cx + d) ln b (ln ((h - k)/a)/ln b) = cx + d  (ln ((h - k)/a)/ln b)  -  d  =  cx  ((ln ((h - k)/a)/ln b) - d)/c  =  x  (((ln ((h - k)/a)))/ln b) - d)/c  = ...

= (1/cos x   +    sin x/cos x) dx = (1 + sin x)/cos x   dx = (1 + sin x)(1 - sin x)/(cos x)(1 - sin x)   dx = (1 - (sin x)^2)/(cos x)(1 - sin x)  dx = (cos x)^2/(cos x)(1 - sin x)  dx  =...

Major radius = 45 Minor radius = 20 (x^2)/(45)^(2)  +  (y^2)/(20)^(2)  =  1

Isolate x on one side and y on the other. (2x dx)/(x^2  +  1)   =  dy/y Integrate ln (x^2  +  1)  = ln |y| +  c ln (1  +  1)  =  ln 2  +  c ln 2  =  ln 2  + ...