Sr Examen
Integral de (1/pi)*sin((x/2)^2)*sin(x) dx
Solución
Ha introducido
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2*pi / | | / 2\ | |/x\ | | sin||-| | | \\2/ / | ---------*sin(x) dx | pi | / 0
$$\int\limits_{0}^{2 \pi} \frac{\sin{\left(\left(\frac{x}{2}\right)^{2} \right)}}{\pi} \sin{\left(x \right)}\, dx$$
Integral((sin((x/2)^2)/pi)*sin(x), (x, 0, 2*pi))
Respuesta (Indefinida)
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/ / ___ / x\\ / ___ / x\\ \ / / ___ / x\\ / ___ / x\\ \
| |\/ 2 *|1 + -|| |\/ 2 *|1 + -|| | | |\/ 2 *|1 - -|| |\/ 2 *|1 - -|| |
/ ___ ____ | | \ 2/| | \ 2/| | ___ ____ | | \ 2/| | \ 2/| |
| \/ 2 *\/ pi *|cos(1)*C|-------------| + S|-------------|*sin(1)| I*\/ 2 *\/ pi *|- I*cos(1)*C|-------------| - I*S|-------------|*sin(1)|
| / 2\ | | ____ | | ____ | | | | ____ | | ____ | |
| |/x\ | \ \ \/ pi / \ \/ pi / / \ \ \/ pi / \ \/ pi / /
| sin||-| | - ---------------------------------------------------------------- + ------------------------------------------------------------------------
| \\2/ / 2 2
| ---------*sin(x) dx = C + ---------------------------------------------------------------------------------------------------------------------------------------------
| pi pi
|
/
$$\int \frac{\sin{\left(\left(\frac{x}{2}\right)^{2} \right)}}{\pi} \sin{\left(x \right)}\, dx = C + \frac{- \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(1 \right)} C\left(\frac{\sqrt{2} \left(\frac{x}{2} + 1\right)}{\sqrt{\pi}}\right) + \sin{\left(1 \right)} S\left(\frac{\sqrt{2} \left(\frac{x}{2} + 1\right)}{\sqrt{\pi}}\right)\right)}{2} + \frac{\sqrt{2} i \sqrt{\pi} \left(- i \cos{\left(1 \right)} C\left(\frac{\sqrt{2} \left(1 - \frac{x}{2}\right)}{\sqrt{\pi}}\right) - i \sin{\left(1 \right)} S\left(\frac{\sqrt{2} \left(1 - \frac{x}{2}\right)}{\sqrt{\pi}}\right)\right)}{2}}{\pi}$$
Gráfica
Respuesta
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/ / ___ \ / ___ \ \ / / ___ \ / ___ \ \ / / ___ \ / ___ \ \ / / ___ \ / ___ \ \
___ ____ | |\/ 2 *(-1 + pi)| |\/ 2 *(-1 + pi)| | ___ ____ | |\/ 2 *(1 + pi)| |\/ 2 *(1 + pi)| | ___ ____ | |\/ 2 | |\/ 2 | | ___ ____ | |\/ 2 | |\/ 2 | |
\/ 2 *\/ pi *|cos(1)*C|---------------| + S|---------------|*sin(1)| \/ 2 *\/ pi *|cos(1)*C|--------------| + S|--------------|*sin(1)| \/ 2 *\/ pi *|- cos(1)*C|------| - S|------|*sin(1)| \/ 2 *\/ pi *|cos(1)*C|------| + S|------|*sin(1)|
| | ____ | | ____ | | | | ____ | | ____ | | | | ____| | ____| | | | ____| | ____| |
\ \ \/ pi / \ \/ pi / / \ \ \/ pi / \ \/ pi / / \ \\/ pi / \\/ pi / / \ \\/ pi / \\/ pi / /
-------------------------------------------------------------------- - ------------------------------------------------------------------ ---------------------------------------------------- - --------------------------------------------------
2 2 2 2
----------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------
pi pi
$$\frac{- \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(1 \right)} C\left(\frac{\sqrt{2} \left(1 + \pi\right)}{\sqrt{\pi}}\right) + \sin{\left(1 \right)} S\left(\frac{\sqrt{2} \left(1 + \pi\right)}{\sqrt{\pi}}\right)\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(1 \right)} C\left(\frac{\sqrt{2} \left(-1 + \pi\right)}{\sqrt{\pi}}\right) + \sin{\left(1 \right)} S\left(\frac{\sqrt{2} \left(-1 + \pi\right)}{\sqrt{\pi}}\right)\right)}{2}}{\pi} - \frac{\frac{\sqrt{2} \sqrt{\pi} \left(- \cos{\left(1 \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) - \sin{\left(1 \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2} - \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(1 \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) + \cos{\left(1 \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2}}{\pi}$$
=
=
/ / ___ \ / ___ \ \ / / ___ \ / ___ \ \ / / ___ \ / ___ \ \ / / ___ \ / ___ \ \
___ ____ | |\/ 2 *(-1 + pi)| |\/ 2 *(-1 + pi)| | ___ ____ | |\/ 2 *(1 + pi)| |\/ 2 *(1 + pi)| | ___ ____ | |\/ 2 | |\/ 2 | | ___ ____ | |\/ 2 | |\/ 2 | |
\/ 2 *\/ pi *|cos(1)*C|---------------| + S|---------------|*sin(1)| \/ 2 *\/ pi *|cos(1)*C|--------------| + S|--------------|*sin(1)| \/ 2 *\/ pi *|- cos(1)*C|------| - S|------|*sin(1)| \/ 2 *\/ pi *|cos(1)*C|------| + S|------|*sin(1)|
| | ____ | | ____ | | | | ____ | | ____ | | | | ____| | ____| | | | ____| | ____| |
\ \ \/ pi / \ \/ pi / / \ \ \/ pi / \ \/ pi / / \ \\/ pi / \\/ pi / / \ \\/ pi / \\/ pi / /
-------------------------------------------------------------------- - ------------------------------------------------------------------ ---------------------------------------------------- - --------------------------------------------------
2 2 2 2
----------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------
pi pi
$$\frac{- \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(1 \right)} C\left(\frac{\sqrt{2} \left(1 + \pi\right)}{\sqrt{\pi}}\right) + \sin{\left(1 \right)} S\left(\frac{\sqrt{2} \left(1 + \pi\right)}{\sqrt{\pi}}\right)\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(1 \right)} C\left(\frac{\sqrt{2} \left(-1 + \pi\right)}{\sqrt{\pi}}\right) + \sin{\left(1 \right)} S\left(\frac{\sqrt{2} \left(-1 + \pi\right)}{\sqrt{\pi}}\right)\right)}{2}}{\pi} - \frac{\frac{\sqrt{2} \sqrt{\pi} \left(- \cos{\left(1 \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) - \sin{\left(1 \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2} - \frac{\sqrt{2} \sqrt{\pi} \left(\sin{\left(1 \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) + \cos{\left(1 \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2}}{\pi}$$
(sqrt(2)*sqrt(pi)*(cos(1)*fresnelc(sqrt(2)*(-1 + pi)/sqrt(pi)) + fresnels(sqrt(2)*(-1 + pi)/sqrt(pi))*sin(1))/2 - sqrt(2)*sqrt(pi)*(cos(1)*fresnelc(sqrt(2)*(1 + pi)/sqrt(pi)) + fresnels(sqrt(2)*(1 + pi)/sqrt(pi))*sin(1))/2)/pi - (sqrt(2)*sqrt(pi)*(-cos(1)*fresnelc(sqrt(2)/sqrt(pi)) - fresnels(sqrt(2)/sqrt(pi))*sin(1))/2 - sqrt(2)*sqrt(pi)*(cos(1)*fresnelc(sqrt(2)/sqrt(pi)) + fresnels(sqrt(2)/sqrt(pi))*sin(1))/2)/pi
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.