Sr Examen
Integral de (a*sin^3(x/3))^2 dx
Solución
Ha introducido
[src]
pi / | | 2 | / 3/x\\ | |a*sin |-|| dx | \ \3// | / pi -- 2
$$\int\limits_{\frac{\pi}{2}}^{\pi} \left(a \sin^{3}{\left(\frac{x}{3} \right)}\right)^{2}\, dx$$
Integral((a*sin(x/3)^3)^2, (x, pi/2, pi))
Respuesta (Indefinida)
[src]
/ | 2 5/x\ 2 3/x\ 2 /x\ 2 11/x\ 2 9/x\ 2 7/x\ 2 12/x\ 2 2/x\ 2 10/x\ 2 4/x\ 2 8/x\ 2 6/x\ | 2 396*a *tan |-| 170*a *tan |-| 30*a *tan|-| 2 30*a *tan |-| 170*a *tan |-| 396*a *tan |-| 5*x*a *tan |-| 30*x*a *tan |-| 30*x*a *tan |-| 75*x*a *tan |-| 75*x*a *tan |-| 100*x*a *tan |-| | / 3/x\\ \6/ \6/ \6/ 5*x*a \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ | |a*sin |-|| dx = C - ------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- | \ \3// 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ 12/x\ 2/x\ 10/x\ 4/x\ 8/x\ 6/x\ | 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| 16 + 16*tan |-| + 96*tan |-| + 96*tan |-| + 240*tan |-| + 240*tan |-| + 320*tan |-| / \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/ \6/
$$\int \left(a \sin^{3}{\left(\frac{x}{3} \right)}\right)^{2}\, dx = C + \frac{5 a^{2} x \tan^{12}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{30 a^{2} x \tan^{10}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{75 a^{2} x \tan^{8}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{100 a^{2} x \tan^{6}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{75 a^{2} x \tan^{4}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{30 a^{2} x \tan^{2}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{5 a^{2} x}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{30 a^{2} \tan^{11}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{170 a^{2} \tan^{9}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{396 a^{2} \tan^{7}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} - \frac{396 a^{2} \tan^{5}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} - \frac{170 a^{2} \tan^{3}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} - \frac{30 a^{2} \tan{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16}$$
Respuesta
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/ ___ \ / ___ \ 2 | 27*\/ 3 5*pi| 2 | 9*\/ 3 5*pi| a *|- -------- + ----| - a *|- ------- + ----| \ 64 16 / \ 32 32 /
$$- a^{2} \left(- \frac{9 \sqrt{3}}{32} + \frac{5 \pi}{32}\right) + a^{2} \left(- \frac{27 \sqrt{3}}{64} + \frac{5 \pi}{16}\right)$$
=
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/ ___ \ / ___ \ 2 | 27*\/ 3 5*pi| 2 | 9*\/ 3 5*pi| a *|- -------- + ----| - a *|- ------- + ----| \ 64 16 / \ 32 32 /
$$- a^{2} \left(- \frac{9 \sqrt{3}}{32} + \frac{5 \pi}{32}\right) + a^{2} \left(- \frac{27 \sqrt{3}}{64} + \frac{5 \pi}{16}\right)$$
a^2*(-27*sqrt(3)/64 + 5*pi/16) - a^2*(-9*sqrt(3)/32 + 5*pi/32)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.