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Resolución de integrales/ sin^5/ xcosx/ (sin^5)xcosx

Integral de (sin^5)xcosx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                    
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 |     5               
 |  sin (x)*x*cos(x) dx
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0
∫01xsin⁡5(x)cos⁡(x) dx\int\limits_{0}^{1} x \sin^{5}{\left(x \right)} \cos{\left(x \right)}\, dx 
Integral((sin(x)^5*x)*cos(x), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                           
 |                                  6           3       3           5                     6            5                    2       4             4       2   
 |    5                      5*x*cos (x)   5*cos (x)*sin (x)   5*cos (x)*sin(x)   11*x*sin (x)   11*sin (x)*cos(x)   5*x*cos (x)*sin (x)   5*x*cos (x)*sin (x)
 | sin (x)*x*cos(x) dx = C - ----------- + ----------------- + ---------------- + ------------ + ----------------- - ------------------- - -------------------
 |                                96               36                 96               96                96                   32                    32        
/
∫xsin⁡5(x)cos⁡(x) dx=C+11xsin⁡6(x)96−5xsin⁡4(x)cos⁡2(x)32−5xsin⁡2(x)cos⁡4(x)32−5xcos⁡6(x)96+11sin⁡5(x)cos⁡(x)96+5sin⁡3(x)cos⁡3(x)36+5sin⁡(x)cos⁡5(x)96\int x \sin^{5}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{11 x \sin^{6}{\left(x \right)}}{96} - \frac{5 x \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{32} - \frac{5 x \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{32} - \frac{5 x \cos^{6}{\left(x \right)}}{96} + \frac{11 \sin^{5}{\left(x \right)} \cos{\left(x \right)}}{96} + \frac{5 \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{36} + \frac{5 \sin{\left(x \right)} \cos^{5}{\left(x \right)}}{96}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.000.25
Trazar el gráfico f(x)
Respuesta [src] 
       6            6           2       4           4       2           3       3           5                   5          
  5*cos (1)   11*sin (1)   5*cos (1)*sin (1)   5*cos (1)*sin (1)   5*cos (1)*sin (1)   5*cos (1)*sin(1)   11*sin (1)*cos(1)
- --------- + ---------- - ----------------- - ----------------- + ----------------- + ---------------- + -----------------
      96          96               32                  32                  36                 96                  96
−5sin⁡4(1)cos⁡2(1)32−5sin⁡2(1)cos⁡4(1)32−5cos⁡6(1)96+5sin⁡(1)cos⁡5(1)96+5sin⁡3(1)cos⁡3(1)36+11sin⁡5(1)cos⁡(1)96+11sin⁡6(1)96- \frac{5 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} - \frac{5 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{32} - \frac{5 \cos^{6}{\left(1 \right)}}{96} + \frac{5 \sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{96} + \frac{5 \sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{36} + \frac{11 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{96} + \frac{11 \sin^{6}{\left(1 \right)}}{96} 
=
= 
       6            6           2       4           4       2           3       3           5                   5          
  5*cos (1)   11*sin (1)   5*cos (1)*sin (1)   5*cos (1)*sin (1)   5*cos (1)*sin (1)   5*cos (1)*sin(1)   11*sin (1)*cos(1)
- --------- + ---------- - ----------------- - ----------------- + ----------------- + ---------------- + -----------------
      96          96               32                  32                  36                 96                  96
−5sin⁡4(1)cos⁡2(1)32−5sin⁡2(1)cos⁡4(1)32−5cos⁡6(1)96+5sin⁡(1)cos⁡5(1)96+5sin⁡3(1)cos⁡3(1)36+11sin⁡5(1)cos⁡(1)96+11sin⁡6(1)96- \frac{5 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} - \frac{5 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{32} - \frac{5 \cos^{6}{\left(1 \right)}}{96} + \frac{5 \sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{96} + \frac{5 \sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{36} + \frac{11 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{96} + \frac{11 \sin^{6}{\left(1 \right)}}{96} 
-5*cos(1)^6/96 + 11*sin(1)^6/96 - 5*cos(1)^2*sin(1)^4/32 - 5*cos(1)^4*sin(1)^2/32 + 5*cos(1)^3*sin(1)^3/36 + 5*cos(1)^5*sin(1)/96 + 11*sin(1)^5*cos(1)/96
Respuesta numérica [src] 
0.0482736235980718
0.0482736235980718

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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