Sr Examen
Integral de ((sin^4)(t))*((cos^2)(t)) dt
Solución
Ha introducido
[src]
13*pi
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6
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11*I*p
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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 /
$$\int t \sin^{4}{\left(t \right)} t \cos^{2}{\left(t \right)}\, dt = C + \frac{t^{3} \sin^{6}{\left(t \right)}}{48} + \frac{t^{3} \sin^{4}{\left(t \right)} \cos^{2}{\left(t \right)}}{16} + \frac{t^{3} \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}}{16} + \frac{t^{3} \cos^{6}{\left(t \right)}}{48} + \frac{t^{2} \sin^{5}{\left(t \right)} \cos{\left(t \right)}}{16} - \frac{t^{2} \sin^{3}{\left(t \right)} \cos^{3}{\left(t \right)}}{6} - \frac{t^{2} \sin{\left(t \right)} \cos^{5}{\left(t \right)}}{16} + \frac{7 t \sin^{6}{\left(t \right)}}{1152} + \frac{31 t \sin^{4}{\left(t \right)} \cos^{2}{\left(t \right)}}{384} - \frac{t \sin^{2}{\left(t \right)} \cos^{4}{\left(t \right)}}{384} - \frac{25 t \cos^{6}{\left(t \right)}}{1152} + \frac{7 \sin^{5}{\left(t \right)} \cos{\left(t \right)}}{1152} + \frac{\sin^{3}{\left(t \right)} \cos^{3}{\left(t \right)}}{27} + \frac{25 \sin{\left(t \right)} \cos^{5}{\left(t \right)}}{1152}$$
Respuesta
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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
$$- \frac{1331 i p^{3} \sinh^{6}{\left(\frac{11 p}{6} \right)}}{10368} + \frac{1331 i p^{3} \sinh^{4}{\left(\frac{11 p}{6} \right)} \cosh^{2}{\left(\frac{11 p}{6} \right)}}{3456} - \frac{1331 i p^{3} \sinh^{2}{\left(\frac{11 p}{6} \right)} \cosh^{4}{\left(\frac{11 p}{6} \right)}}{3456} + \frac{1331 i p^{3} \cosh^{6}{\left(\frac{11 p}{6} \right)}}{10368} + \frac{121 i p^{2} \sinh^{5}{\left(\frac{11 p}{6} \right)} \cosh{\left(\frac{11 p}{6} \right)}}{576} + \frac{121 i p^{2} \sinh^{3}{\left(\frac{11 p}{6} \right)} \cosh^{3}{\left(\frac{11 p}{6} \right)}}{216} - \frac{121 i p^{2} \sinh{\left(\frac{11 p}{6} \right)} \cosh^{5}{\left(\frac{11 p}{6} \right)}}{576} + \frac{77 i p \sinh^{6}{\left(\frac{11 p}{6} \right)}}{6912} - \frac{341 i p \sinh^{4}{\left(\frac{11 p}{6} \right)} \cosh^{2}{\left(\frac{11 p}{6} \right)}}{2304} - \frac{11 i p \sinh^{2}{\left(\frac{11 p}{6} \right)} \cosh^{4}{\left(\frac{11 p}{6} \right)}}{2304} + \frac{275 i p \cosh^{6}{\left(\frac{11 p}{6} \right)}}{6912} - \frac{7 i \sinh^{5}{\left(\frac{11 p}{6} \right)} \cosh{\left(\frac{11 p}{6} \right)}}{1152} + \frac{i \sinh^{3}{\left(\frac{11 p}{6} \right)} \cosh^{3}{\left(\frac{11 p}{6} \right)}}{27} - \frac{25 i \sinh{\left(\frac{11 p}{6} \right)} \cosh^{5}{\left(\frac{11 p}{6} \right)}}{1152} - \frac{169 \sqrt{3} \pi^{2}}{2304} - \frac{169 \pi}{13824} + \frac{5 \sqrt{3}}{1024} + \frac{2197 \pi^{3}}{10368}$$
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=
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.