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Resolución de integrales/ cos^2/ sin^4/ ((sin^4)(t))*((cos^2)(t))

Integral de ((sin^4)(t))*((cos^2)(t)) dt

Solución

Ha introducido [src]

  13*pi                      
  -----                      
    6                        
    /                        
   |                         
   |      4         2        
   |   sin (t)*t*cos (t)*t dt
   |                         
  /                          
11*I*p                       
------                       
  6

\int\limits_{\frac{11 i p}{6}}^{\frac{13 \pi}{6}} t \sin^{4}{\left(t \right)} t \cos^{2}{\left(t \right)}\, dt

Integral((sin(t)^4*t)*(cos(t)^2*t), (t, 11*i*p/6, 13*pi/6))

Respuesta (Indefinida) [src]

  /                                                                                                                                                                                                                                                                                            
 |                                      6         3       3       3    6       3    6             6           5                   5              2    3       3       2    5                  4       2       2    5              3    2       4       3    4       2              2       4   
 |    4         2               25*t*cos (t)   cos (t)*sin (t)   t *cos (t)   t *sin (t)   7*t*sin (t)   7*sin (t)*cos(t)   25*cos (t)*sin(t)   t *cos (t)*sin (t)   t *cos (t)*sin(t)   t*cos (t)*sin (t)   t *sin (t)*cos(t)   t *cos (t)*sin (t)   t *cos (t)*sin (t)   31*t*cos (t)*sin (t)
 | sin (t)*t*cos (t)*t dt = C - ------------ + --------------- + ---------- + ---------- + ----------- + ---------------- + ----------------- - ------------------ - ----------------- - ----------------- + ----------------- + ------------------ + ------------------ + --------------------
 |                                  1152              27             48           48           1152            1152                1152                 6                    16                 384                  16                  16                   16                   384         
/

\int t \sin^{4}{\left(t \right)} t \cos^{2}{\left(t \right)}\, dt = C + \frac{t^{3} \sin^{6}{\left(t \right)}}{48} + \frac{t^{3} \sin^{4}{\left(t \right)} \cos^{2}{\left(t \right)}}{16} + \frac{t^{3} \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}}{16} + \frac{t^{3} \cos^{6}{\left(t \right)}}{48} + \frac{t^{2} \sin^{5}{\left(t \right)} \cos{\left(t \right)}}{16} - \frac{t^{2} \sin^{3}{\left(t \right)} \cos^{3}{\left(t \right)}}{6} - \frac{t^{2} \sin{\left(t \right)} \cos^{5}{\left(t \right)}}{16} + \frac{7 t \sin^{6}{\left(t \right)}}{1152} + \frac{31 t \sin^{4}{\left(t \right)} \cos^{2}{\left(t \right)}}{384} - \frac{t \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}}{384} - \frac{25 t \cos^{6}{\left(t \right)}}{1152} + \frac{7 \sin^{5}{\left(t \right)} \cos{\left(t \right)}}{1152} + \frac{\sin^{3}{\left(t \right)} \cos^{3}{\left(t \right)}}{27} + \frac{25 \sin{\left(t \right)} \cos^{5}{\left(t \right)}}{1152}

Simplificar

Respuesta [src]

                                                        3     6/11*p\            5/11*p\     /11*p\           5/11*p\     /11*p\         3/11*p\     3/11*p\              6/11*p\               6/11*p\           3     6/11*p\           3     4/11*p\     2/11*p\               2/11*p\     4/11*p\          2     5/11*p\     /11*p\              4/11*p\     2/11*p\          2     3/11*p\     3/11*p\          2     5/11*p\     /11*p\           3     2/11*p\     4/11*p\
               ___          3         ___   2   1331*I*p *sinh |----|   25*I*cosh |----|*sinh|----|   7*I*sinh |----|*cosh|----|   I*cosh |----|*sinh |----|   77*I*p*sinh |----|   275*I*p*cosh |----|   1331*I*p *cosh |----|   1331*I*p *cosh |----|*sinh |----|   341*I*p*cosh |----|*sinh |----|   121*I*p *cosh |----|*sinh|----|   11*I*p*cosh |----|*sinh |----|   121*I*p *cosh |----|*sinh |----|   121*I*p *sinh |----|*cosh|----|   1331*I*p *cosh |----|*sinh |----|
  169*pi   5*\/ 3    2197*pi    169*\/ 3 *pi                   \ 6  /             \ 6  /     \ 6  /            \ 6  /     \ 6  /          \ 6  /      \ 6  /               \ 6  /                \ 6  /                  \ 6  /                  \ 6  /      \ 6  /                \ 6  /      \ 6  /                 \ 6  /     \ 6  /               \ 6  /      \ 6  /                 \ 6  /      \ 6  /                 \ 6  /     \ 6  /                  \ 6  /      \ 6  /
- ------ + ------- + -------- - ------------- - --------------------- - --------------------------- - -------------------------- + ------------------------- + ------------------ + ------------------- + --------------------- - --------------------------------- - ------------------------------- - ------------------------------- - ------------------------------ + -------------------------------- + ------------------------------- + ---------------------------------
  13824      1024     10368          2304               10368                       1152                         1152                          27                     6912                  6912                  10368                          3456                               2304                              576                              2304                              216                                576                                3456

- \frac{1331 i p^{3} \sinh^{6}{\left(\frac{11 p}{6} \right)}}{10368} + \frac{1331 i p^{3} \sinh^{4}{\left(\frac{11 p}{6} \right)} \cosh^{2}{\left(\frac{11 p}{6} \right)}}{3456} - \frac{1331 i p^{3} \sinh^{2}{\left(\frac{11 p}{6} \right)} \cosh^{4}{\left(\frac{11 p}{6} \right)}}{3456} + \frac{1331 i p^{3} \cosh^{6}{\left(\frac{11 p}{6} \right)}}{10368} + \frac{121 i p^{2} \sinh^{5}{\left(\frac{11 p}{6} \right)} \cosh{\left(\frac{11 p}{6} \right)}}{576} + \frac{121 i p^{2} \sinh^{3}{\left(\frac{11 p}{6} \right)} \cosh^{3}{\left(\frac{11 p}{6} \right)}}{216} - \frac{121 i p^{2} \sinh{\left(\frac{11 p}{6} \right)} \cosh^{5}{\left(\frac{11 p}{6} \right)}}{576} + \frac{77 i p \sinh^{6}{\left(\frac{11 p}{6} \right)}}{6912} - \frac{341 i p \sinh^{4}{\left(\frac{11 p}{6} \right)} \cosh^{2}{\left(\frac{11 p}{6} \right)}}{2304} - \frac{11 i p \sinh^{2}{\left(\frac{11 p}{6} \right)} \cosh^{4}{\left(\frac{11 p}{6} \right)}}{2304} + \frac{275 i p \cosh^{6}{\left(\frac{11 p}{6} \right)}}{6912} - \frac{7 i \sinh^{5}{\left(\frac{11 p}{6} \right)} \cosh{\left(\frac{11 p}{6} \right)}}{1152} + \frac{i \sinh^{3}{\left(\frac{11 p}{6} \right)} \cosh^{3}{\left(\frac{11 p}{6} \right)}}{27} - \frac{25 i \sinh{\left(\frac{11 p}{6} \right)} \cosh^{5}{\left(\frac{11 p}{6} \right)}}{1152} - \frac{169 \sqrt{3} \pi^{2}}{2304} - \frac{169 \pi}{13824} + \frac{5 \sqrt{3}}{1024} + \frac{2197 \pi^{3}}{10368}

                                                        3     6/11*p\            5/11*p\     /11*p\           5/11*p\     /11*p\         3/11*p\     3/11*p\              6/11*p\               6/11*p\           3     6/11*p\           3     4/11*p\     2/11*p\               2/11*p\     4/11*p\          2     5/11*p\     /11*p\              4/11*p\     2/11*p\          2     3/11*p\     3/11*p\          2     5/11*p\     /11*p\           3     2/11*p\     4/11*p\
               ___          3         ___   2   1331*I*p *sinh |----|   25*I*cosh |----|*sinh|----|   7*I*sinh |----|*cosh|----|   I*cosh |----|*sinh |----|   77*I*p*sinh |----|   275*I*p*cosh |----|   1331*I*p *cosh |----|   1331*I*p *cosh |----|*sinh |----|   341*I*p*cosh |----|*sinh |----|   121*I*p *cosh |----|*sinh|----|   11*I*p*cosh |----|*sinh |----|   121*I*p *cosh |----|*sinh |----|   121*I*p *sinh |----|*cosh|----|   1331*I*p *cosh |----|*sinh |----|
  169*pi   5*\/ 3    2197*pi    169*\/ 3 *pi                   \ 6  /             \ 6  /     \ 6  /            \ 6  /     \ 6  /          \ 6  /      \ 6  /               \ 6  /                \ 6  /                  \ 6  /                  \ 6  /      \ 6  /                \ 6  /      \ 6  /                 \ 6  /     \ 6  /               \ 6  /      \ 6  /                 \ 6  /      \ 6  /                 \ 6  /     \ 6  /                  \ 6  /      \ 6  /
- ------ + ------- + -------- - ------------- - --------------------- - --------------------------- - -------------------------- + ------------------------- + ------------------ + ------------------- + --------------------- - --------------------------------- - ------------------------------- - ------------------------------- - ------------------------------ + -------------------------------- + ------------------------------- + ---------------------------------
  13824      1024     10368          2304               10368                       1152                         1152                          27                     6912                  6912                  10368                          3456                               2304                              576                              2304                              216                                576                                3456

- \frac{1331 i p^{3} \sinh^{6}{\left(\frac{11 p}{6} \right)}}{10368} + \frac{1331 i p^{3} \sinh^{4}{\left(\frac{11 p}{6} \right)} \cosh^{2}{\left(\frac{11 p}{6} \right)}}{3456} - \frac{1331 i p^{3} \sinh^{2}{\left(\frac{11 p}{6} \right)} \cosh^{4}{\left(\frac{11 p}{6} \right)}}{3456} + \frac{1331 i p^{3} \cosh^{6}{\left(\frac{11 p}{6} \right)}}{10368} + \frac{121 i p^{2} \sinh^{5}{\left(\frac{11 p}{6} \right)} \cosh{\left(\frac{11 p}{6} \right)}}{576} + \frac{121 i p^{2} \sinh^{3}{\left(\frac{11 p}{6} \right)} \cosh^{3}{\left(\frac{11 p}{6} \right)}}{216} - \frac{121 i p^{2} \sinh{\left(\frac{11 p}{6} \right)} \cosh^{5}{\left(\frac{11 p}{6} \right)}}{576} + \frac{77 i p \sinh^{6}{\left(\frac{11 p}{6} \right)}}{6912} - \frac{341 i p \sinh^{4}{\left(\frac{11 p}{6} \right)} \cosh^{2}{\left(\frac{11 p}{6} \right)}}{2304} - \frac{11 i p \sinh^{2}{\left(\frac{11 p}{6} \right)} \cosh^{4}{\left(\frac{11 p}{6} \right)}}{2304} + \frac{275 i p \cosh^{6}{\left(\frac{11 p}{6} \right)}}{6912} - \frac{7 i \sinh^{5}{\left(\frac{11 p}{6} \right)} \cosh{\left(\frac{11 p}{6} \right)}}{1152} + \frac{i \sinh^{3}{\left(\frac{11 p}{6} \right)} \cosh^{3}{\left(\frac{11 p}{6} \right)}}{27} - \frac{25 i \sinh{\left(\frac{11 p}{6} \right)} \cosh^{5}{\left(\frac{11 p}{6} \right)}}{1152} - \frac{169 \sqrt{3} \pi^{2}}{2304} - \frac{169 \pi}{13824} + \frac{5 \sqrt{3}}{1024} + \frac{2197 \pi^{3}}{10368}

-169*pi/13824 + 5*sqrt(3)/1024 + 2197*pi^3/10368 - 169*sqrt(3)*pi^2/2304 - 1331*i*p^3*sinh(11*p/6)^6/10368 - 25*i*cosh(11*p/6)^5*sinh(11*p/6)/1152 - 7*i*sinh(11*p/6)^5*cosh(11*p/6)/1152 + i*cosh(11*p/6)^3*sinh(11*p/6)^3/27 + 77*i*p*sinh(11*p/6)^6/6912 + 275*i*p*cosh(11*p/6)^6/6912 + 1331*i*p^3*cosh(11*p/6)^6/10368 - 1331*i*p^3*cosh(11*p/6)^4*sinh(11*p/6)^2/3456 - 341*i*p*cosh(11*p/6)^2*sinh(11*p/6)^4/2304 - 121*i*p^2*cosh(11*p/6)^5*sinh(11*p/6)/576 - 11*i*p*cosh(11*p/6)^4*sinh(11*p/6)^2/2304 + 121*i*p^2*cosh(11*p/6)^3*sinh(11*p/6)^3/216 + 121*i*p^2*sinh(11*p/6)^5*cosh(11*p/6)/576 + 1331*i*p^3*cosh(11*p/6)^2*sinh(11*p/6)^4/3456

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

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Expresiones con funciones

Integral de ((sin^4)(t))*((cos^2)(t)) dt

Límites de integración:

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Definida a trozos:

Solución