Integración por sustitución ejemplo 11


Esto es lo que usted escribira:

cos[x] / (sin[x] + cos[x]^2)

∫ cos(x)/(cos(x)^2 + sin(x)) d x
<br />          Sustitución <br />          s = sin(x), <br />          d s = cos(x) d x . <br />         
= ∫ 1/(-s^2 + s + 1) d s
<br />          Factorizando el denominador en términos lineales . <br />         
= ∫ -1/((s + 1/2 (-1 - 5^(1/2))) (s + 1/2 (-1 + 5^(1/2)))) d s
<br />          Factorizando constantess . <br />         
= -∫ 1/((s + 1/2 (-1 - 5^(1/2))) (s + 1/2 (-1 + 5^(1/2)))) d s
<br />          Separando las fracciones en fracciones parciales . <br />         
= -∫ (1/((1/2 (-1 - 5^(1/2)) + 1/2 (1 - 5^(1/2))) (s + 1/2 (-1 + 5^(1/2)))) + 1/((1/2 (-1 + 5^(1/2)) + 1/2 (1 + 5^(1/2))) (s + 1/2 (-1 - 5^(1/2))))) d s
<br />          Integrando la suma término-por-término y factorizando las constantes . <br />         
= 1/5^(1/2) ∫ 1/(s + 1/2 (-1 + 5^(1/2))) d s - 1/5^(1/2) ∫ 1/(s + 1/2 (-1 - 5^(1/2))) d s
<br />          Para el Integrante 1/(s + 1/2 (-1 - 5^(1/2))), <br />          sustituya t = s + 1/2 (-1 - 5^(1/2)), <br />          d t = 1 d s . <br />         
= 1/5^(1/2) ∫ 1/(s + 1/2 (-1 + 5^(1/2))) d s - 1/5^(1/2) ∫ 1/t d t
<br />          Para el Integrante 1/(s + 1/2 (-1 + 5^(1/2))), <br />          sustituya w = s + 1/2 (-1 + 5^(1/2)), <br />          d w = 1 d s . <br />         
= 1/5^(1/2) ∫ 1/w d w - 1/5^(1/2) ∫ 1/t d t
<br />          La Integral de 1/t es log(t) . <br />         
= 1/5^(1/2) ∫ 1/w d w - log(t)/5^(1/2)
<br />          La Integral de 1/w es log(w) . <br />         
= log(w)/5^(1/2) - log(t)/5^(1/2) + ÷r
<br />          Resustituyendo w = s + 1/2 (-1 + 5^(1/2)) . <br />         
= log(s + 1/2 (-1 + 5^(1/2)))/5^(1/2) - log(t)/5^(1/2) + ÷r
<br />          Resustituyendo t = s + 1/2 (-1 - 5^(1/2)) . <br />         
= log(s + 1/2 (-1 + 5^(1/2)))/5^(1/2) - log(s + 1/2 (-1 - 5^(1/2)))/5^(1/2) + ÷r
<br />          Resustituyendo s = sin(x) . <br />         
= log(sin(x) + 1/2 (-1 + 5^(1/2)))/5^(1/2) - log(sin(x) + 1/2 (-1 - 5^(1/2)))/5^(1/2) + ÷r
<br />          Factor por otra expresión para ver el resultado . <br />         
= -(log(sin(x) + 1/2 (-1 - 5^(1/2))) - log(sin(x) + 1/2 (-1 + 5^(1/2))))/5^(1/2) + ÷r

Converted by Mathematica  (March 3, 2003)