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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...

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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...

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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...

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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.

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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 ...

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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

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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                                      ...

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Find dz/dx? (answer)

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)    

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 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  ...

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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) +...

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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       ...

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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 [...

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= (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  =...

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