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youjnhuhn2

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Posts posted by youjnhuhn2

  1. Buna am si eu o problema daca ma poate ajuta cineva cu ceva calcul/relatii de calcul.

    Calculand sarcina pe care o pot dezvolta diverse baterii (1.5V si 12V), calculati numarul de condensatoare de 1000uF conectate in paralel, necesare stocarii aceleasi sarcini, alimentate la tensiunea inscrisa pe baterie.

    Nu prea stiu ce si cum sa incep :)

  2. Take the integral:

    integral csc(x+1) csc(x+2) dx

    Write csc(x+1) csc(x+2) as 2/(cos(1)-cos(2 x+3)):

    = integral 2/(cos(1)-cos(2 x+3)) dx

    Factor out constants:

    = 2 integral 1/(cos(1)-cos(2 x+3)) dx

    For the integrand 1/(cos(1)-cos(2 x+3)), substitute u = 2 x+3 and du = 2 dx:

    = integral 1/(cos(1)-cos(u)) du

    For the integrand 1/(cos(1)-cos(u)), substitute s = tan(u/2) and ds = 1/2 sec^2(u/2) du. Then transform the integrand using the substitutions sin(u) = (2 s)/(s^2+1), cos(u) = (1-s^2)/(s^2+1) and du = (2 ds)/(s^2+1):

    = integral 2/((s^2+1) (cos(1)-(1-s^2)/(s^2+1))) ds

    Simplify the integrand 2/((s^2+1) (cos(1)-(1-s^2)/(s^2+1))) to get 2/(s^2+s^2 cos(1)-1+cos(1)):

    = integral 2/(s^2+s^2 cos(1)-1+cos(1)) ds

    Factor out constants:

    = 2 integral 1/(s^2+s^2 cos(1)-1+cos(1)) ds

    Factor cos(1)-1 from the denominator:

    = 2 integral 1/((cos(1)-1) ((s^2 (1+cos(1)))/(cos(1)-1)+1)) ds

    Factor out constants:

    = 2/(cos(1)-1) integral 1/((s^2 (1+cos(1)))/(cos(1)-1)+1) ds

    For the integrand 1/((s^2 (1+cos(1)))/(cos(1)-1)+1), substitute p = s cot(1/2) and dp = cot(1/2) ds:

    = (2 tan(1/2))/(cos(1)-1) integral 1/(1-p^2) dp

    The integral of 1/(1-p^2) is tanh^(-1)(p):

    = (2 tan(1/2) tanh^(-1)(p))/(cos(1)-1)+constant

    Substitute back for p = s cot(1/2):

    = csc(1/2) sec(1/2) (-tanh^(-1)(s cot(1/2)))+constant

    Substitute back for s = tan(u/2):

    = csc(1/2) sec(1/2) (-tanh^(-1)(cot(1/2) tan(u/2)))+constant

    Substitute back for u = 2 x+3:

    = csc(1/2) sec(1/2) (-tanh^(-1)(cot(1/2) tan(x+3/2)))+constant

    Which is equivalent for restricted x values to:

    Answer: |

    | = csc(1) (log(sin(x+1))-log(sin(x+2)))+constant

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