Sr Examen
Integral de (sin^22x-1/2) dx
Solución
Ha introducido
[src]
pi -- 2 / | | / 22 1\ | |sin (x) - -| dx | \ 2/ | / pi -- 8
$$\int\limits_{\frac{\pi}{8}}^{\frac{\pi}{2}} \left(\sin^{22}{\left(x \right)} - \frac{1}{2}\right)\, dx$$
Integral(sin(x)^22 - 1/2, (x, pi/8, pi/2))
Gráfica
Respuesta
[src]
21/2 ___________ 19/2 ___________ 17/2 ___________ 13/2 ___________ 15/2 ___________ 9/2 ___________ 11/2 ___________ 7/2 ___________ 3/2 ___________ 5/2 ___________ ___________ ___________
/ ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___ / ___
|1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 / 1 \/ 2 / 1 \/ 2
|- - -----| * / - + ----- 21*|- - -----| * / - + ----- 133*|- - -----| * / - + ----- 323*|- - -----| * / - + ----- 2261*|- - -----| * / - + ----- 4199*|- - -----| * / - + ----- 4199*|- - -----| * / - + ----- 12597*|- - -----| * / - + ----- 29393*|- - -----| * / - + ----- 29393*|- - -----| * / - + ----- 88179* / - - ----- * / - + -----
521895*pi \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \/ 2 4 \/ 2 4
- --------- + -------------------------------- + ----------------------------------- + ------------------------------------ + ------------------------------------ + ------------------------------------- + ------------------------------------ + ------------------------------------- + ------------------------------------- + ------------------------------------- + ------------------------------------- + ---------------------------------------
4194304 22 440 2640 5632 42240 61440 67584 163840 262144 327680 524288
$$- \frac{521895 \pi}{4194304} + \frac{\left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{21}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{22} + \frac{21 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{19}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{440} + \frac{133 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{17}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2640} + \frac{2261 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{15}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{42240} + \frac{323 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{13}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{5632} + \frac{4199 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{11}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{67584} + \frac{4199 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{9}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{61440} + \frac{12597 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{7}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{163840} + \frac{29393 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{5}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{327680} + \frac{29393 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{3}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{262144} + \frac{88179 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{524288}$$
=
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21/2 ___________ 19/2 ___________ 17/2 ___________ 13/2 ___________ 15/2 ___________ 9/2 ___________ 11/2 ___________ 7/2 ___________ 3/2 ___________ 5/2 ___________ ___________ ___________
/ ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___\ / ___ / ___ / ___
|1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | / 1 \/ 2 / 1 \/ 2 / 1 \/ 2
|- - -----| * / - + ----- 21*|- - -----| * / - + ----- 133*|- - -----| * / - + ----- 323*|- - -----| * / - + ----- 2261*|- - -----| * / - + ----- 4199*|- - -----| * / - + ----- 4199*|- - -----| * / - + ----- 12597*|- - -----| * / - + ----- 29393*|- - -----| * / - + ----- 29393*|- - -----| * / - + ----- 88179* / - - ----- * / - + -----
521895*pi \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \2 4 / \/ 2 4 \/ 2 4 \/ 2 4
- --------- + -------------------------------- + ----------------------------------- + ------------------------------------ + ------------------------------------ + ------------------------------------- + ------------------------------------ + ------------------------------------- + ------------------------------------- + ------------------------------------- + ------------------------------------- + ---------------------------------------
4194304 22 440 2640 5632 42240 61440 67584 163840 262144 327680 524288
$$- \frac{521895 \pi}{4194304} + \frac{\left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{21}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{22} + \frac{21 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{19}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{440} + \frac{133 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{17}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2640} + \frac{2261 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{15}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{42240} + \frac{323 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{13}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{5632} + \frac{4199 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{11}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{67584} + \frac{4199 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{9}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{61440} + \frac{12597 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{7}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{163840} + \frac{29393 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{5}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{327680} + \frac{29393 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{3}{2}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{262144} + \frac{88179 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{524288}$$
-521895*pi/4194304 + (1/2 - sqrt(2)/4)^(21/2)*sqrt(1/2 + sqrt(2)/4)/22 + 21*(1/2 - sqrt(2)/4)^(19/2)*sqrt(1/2 + sqrt(2)/4)/440 + 133*(1/2 - sqrt(2)/4)^(17/2)*sqrt(1/2 + sqrt(2)/4)/2640 + 323*(1/2 - sqrt(2)/4)^(13/2)*sqrt(1/2 + sqrt(2)/4)/5632 + 2261*(1/2 - sqrt(2)/4)^(15/2)*sqrt(1/2 + sqrt(2)/4)/42240 + 4199*(1/2 - sqrt(2)/4)^(9/2)*sqrt(1/2 + sqrt(2)/4)/61440 + 4199*(1/2 - sqrt(2)/4)^(11/2)*sqrt(1/2 + sqrt(2)/4)/67584 + 12597*(1/2 - sqrt(2)/4)^(7/2)*sqrt(1/2 + sqrt(2)/4)/163840 + 29393*(1/2 - sqrt(2)/4)^(3/2)*sqrt(1/2 + sqrt(2)/4)/262144 + 29393*(1/2 - sqrt(2)/4)^(5/2)*sqrt(1/2 + sqrt(2)/4)/327680 + 88179*sqrt(1/2 - sqrt(2)/4)*sqrt(1/2 + sqrt(2)/4)/524288
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.