Sr Examen
Integral de 1/(4+3sin(x)^2) dx
Solución
Ha introducido
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1 / | | 1 | ------------- dx | 2 | 4 + 3*sin (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{3 \sin^{2}{\left(x \right)} + 4}\, dx$$
Integral(1/(4 + 3*sin(x)^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ / ___ /x\ \\ / /x pi\ / ___ /x\ \\ / /x pi\ / ___ /x\ \\ / /x pi\ / ___ /x\ \\
| |- - --| | \/ 2 *tan|-| || | |- - --| | \/ 2 *tan|-| || ____________ ____________ | |- - --| | \/ 2 *tan|-| || ____________ ____________ | |- - --| | \/ 2 *tan|-| ||
____ | |2 2 | | \2/ || ___ | |2 2 | | \2/ || ____ / ____ / ____ | |2 2 | | \2/ || ___ / ____ / ____ | |2 2 | | \2/ ||
2*\/ 42 *|pi*floor|------| + atan|---------------|| 10*\/ 2 *|pi*floor|------| + atan|---------------|| \/ 42 *\/ 5 + \/ 21 *\/ 5 - \/ 21 *|pi*floor|------| + atan|---------------|| 5*\/ 2 *\/ 5 + \/ 21 *\/ 5 - \/ 21 *|pi*floor|------| + atan|---------------||
/ | \ pi / | ____________|| | \ pi / | ____________|| | \ pi / | ____________|| | \ pi / | ____________||
| | | / ____ || | | / ____ || | | / ____ || | | / ____ ||
| 1 \ \\/ 5 - \/ 21 // \ \\/ 5 - \/ 21 // \ \\/ 5 + \/ 21 // \ \\/ 5 + \/ 21 //
| ------------- dx = C + --------------------------------------------------- + --------------------------------------------------- + --------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------
| 2 ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________
| 4 + 3*sin (x) / ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____
| 112*\/ 5 - \/ 21 + 24*\/ 21 *\/ 5 - \/ 21 112*\/ 5 - \/ 21 + 24*\/ 21 *\/ 5 - \/ 21 112*\/ 5 - \/ 21 + 24*\/ 21 *\/ 5 - \/ 21 112*\/ 5 - \/ 21 + 24*\/ 21 *\/ 5 - \/ 21
/
$$\int \frac{1}{3 \sin^{2}{\left(x \right)} + 4}\, dx = C + \frac{2 \sqrt{42} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{5 - \sqrt{21}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{24 \sqrt{21} \sqrt{5 - \sqrt{21}} + 112 \sqrt{5 - \sqrt{21}}} + \frac{10 \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{5 - \sqrt{21}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{24 \sqrt{21} \sqrt{5 - \sqrt{21}} + 112 \sqrt{5 - \sqrt{21}}} + \frac{\sqrt{42} \sqrt{5 - \sqrt{21}} \sqrt{\sqrt{21} + 5} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{\sqrt{21} + 5}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{24 \sqrt{21} \sqrt{5 - \sqrt{21}} + 112 \sqrt{5 - \sqrt{21}}} + \frac{5 \sqrt{2} \sqrt{5 - \sqrt{21}} \sqrt{\sqrt{21} + 5} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{\sqrt{21} + 5}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{24 \sqrt{21} \sqrt{5 - \sqrt{21}} + 112 \sqrt{5 - \sqrt{21}}}$$
Gráfica
Respuesta
[src]
/ / ___ \\ ____________ ____________ / / ___ \\
___ | | \/ 2 *tan(1/2)|| ___ / ____ / ____ | | \/ 2 *tan(1/2)||
2*\/ 2 *|-pi + atan|---------------|| \/ 2 *\/ 5 + \/ 21 *\/ 5 - \/ 21 *|-pi + atan|---------------||
| | ____________|| ____________ ____________ | | ____________||
___ | | / ____ || ___ / ____ / ____ | | / ____ ||
2*pi*\/ 2 \ \\/ 5 - \/ 21 // pi*\/ 2 *\/ 5 + \/ 21 *\/ 5 - \/ 21 \ \\/ 5 + \/ 21 //
--------------------------------------------- + --------------------------------------------- + --------------------------------------------- + -------------------------------------------------------------------
____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________
/ ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____
14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21 14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21 14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21 14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21
$$\frac{\sqrt{2} \sqrt{5 - \sqrt{21}} \sqrt{\sqrt{21} + 5} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{\sqrt{21} + 5}} \right)}\right)}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}} + \frac{2 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{5 - \sqrt{21}}} \right)}\right)}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}} + \frac{2 \sqrt{2} \pi}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}} + \frac{\sqrt{2} \pi \sqrt{5 - \sqrt{21}} \sqrt{\sqrt{21} + 5}}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}}$$
=
=
/ / ___ \\ ____________ ____________ / / ___ \\
___ | | \/ 2 *tan(1/2)|| ___ / ____ / ____ | | \/ 2 *tan(1/2)||
2*\/ 2 *|-pi + atan|---------------|| \/ 2 *\/ 5 + \/ 21 *\/ 5 - \/ 21 *|-pi + atan|---------------||
| | ____________|| ____________ ____________ | | ____________||
___ | | / ____ || ___ / ____ / ____ | | / ____ ||
2*pi*\/ 2 \ \\/ 5 - \/ 21 // pi*\/ 2 *\/ 5 + \/ 21 *\/ 5 - \/ 21 \ \\/ 5 + \/ 21 //
--------------------------------------------- + --------------------------------------------- + --------------------------------------------- + -------------------------------------------------------------------
____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________
/ ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____
14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21 14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21 14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21 14*\/ 5 - \/ 21 + 2*\/ 21 *\/ 5 - \/ 21
$$\frac{\sqrt{2} \sqrt{5 - \sqrt{21}} \sqrt{\sqrt{21} + 5} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{\sqrt{21} + 5}} \right)}\right)}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}} + \frac{2 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{5 - \sqrt{21}}} \right)}\right)}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}} + \frac{2 \sqrt{2} \pi}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}} + \frac{\sqrt{2} \pi \sqrt{5 - \sqrt{21}} \sqrt{\sqrt{21} + 5}}{2 \sqrt{21} \sqrt{5 - \sqrt{21}} + 14 \sqrt{5 - \sqrt{21}}}$$
2*pi*sqrt(2)/(14*sqrt(5 - sqrt(21)) + 2*sqrt(21)*sqrt(5 - sqrt(21))) + 2*sqrt(2)*(-pi + atan(sqrt(2)*tan(1/2)/sqrt(5 - sqrt(21))))/(14*sqrt(5 - sqrt(21)) + 2*sqrt(21)*sqrt(5 - sqrt(21))) + pi*sqrt(2)*sqrt(5 + sqrt(21))*sqrt(5 - sqrt(21))/(14*sqrt(5 - sqrt(21)) + 2*sqrt(21)*sqrt(5 - sqrt(21))) + sqrt(2)*sqrt(5 + sqrt(21))*sqrt(5 - sqrt(21))*(-pi + atan(sqrt(2)*tan(1/2)/sqrt(5 + sqrt(21))))/(14*sqrt(5 - sqrt(21)) + 2*sqrt(21)*sqrt(5 - sqrt(21)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.