Sr Examen
Integral de 1/(4+sen^2(x)) dx
Solución
Ha introducido
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1 / | | 1 | ----------- dx | 2 | 4 + sin (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin^{2}{\left(x \right)} + 4}\, dx$$
Integral(1/(4 + 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*\/ 10 *|pi*floor|------| + atan|--------------|| 6*\/ 2 *|pi*floor|------| + atan|--------------|| \/ 10 *\/ 3 + \/ 5 *\/ 3 - \/ 5 *|pi*floor|------| + atan|--------------|| 3*\/ 2 *\/ 3 + \/ 5 *\/ 3 - \/ 5 *|pi*floor|------| + atan|--------------||
/ | \ pi / | ___________|| | \ pi / | ___________|| | \ pi / | ___________|| | \ pi / | ___________||
| | | / ___ || | | / ___ || | | / ___ || | | / ___ ||
| 1 \ \\/ 3 - \/ 5 // \ \\/ 3 - \/ 5 // \ \\/ 3 + \/ 5 // \ \\/ 3 + \/ 5 //
| ----------- dx = C + -------------------------------------------------- + ------------------------------------------------- + ------------------------------------------------------------------------------ + -------------------------------------------------------------------------------
| 2 ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
| 4 + sin (x) / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
| 40*\/ 3 - \/ 5 + 16*\/ 5 *\/ 3 - \/ 5 40*\/ 3 - \/ 5 + 16*\/ 5 *\/ 3 - \/ 5 40*\/ 3 - \/ 5 + 16*\/ 5 *\/ 3 - \/ 5 40*\/ 3 - \/ 5 + 16*\/ 5 *\/ 3 - \/ 5
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 4}\, dx = C + \frac{2 \sqrt{10} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - \sqrt{5}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{16 \sqrt{5} \sqrt{3 - \sqrt{5}} + 40 \sqrt{3 - \sqrt{5}}} + \frac{6 \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - \sqrt{5}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{16 \sqrt{5} \sqrt{3 - \sqrt{5}} + 40 \sqrt{3 - \sqrt{5}}} + \frac{\sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{\sqrt{5} + 3} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{\sqrt{5} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{16 \sqrt{5} \sqrt{3 - \sqrt{5}} + 40 \sqrt{3 - \sqrt{5}}} + \frac{3 \sqrt{2} \sqrt{3 - \sqrt{5}} \sqrt{\sqrt{5} + 3} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{\sqrt{5} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{16 \sqrt{5} \sqrt{3 - \sqrt{5}} + 40 \sqrt{3 - \sqrt{5}}}$$
Gráfica
Respuesta
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/ / ___ \\ ___________ ___________ / / ___ \\
___ | |\/ 2 *tan(1/2)|| ___ / ___ / ___ | |\/ 2 *tan(1/2)||
2*\/ 2 *|-pi + atan|--------------|| \/ 2 *\/ 3 + \/ 5 *\/ 3 - \/ 5 *|-pi + atan|--------------||
| | ___________|| ___________ ___________ | | ___________||
___ | | / ___ || ___ / ___ / ___ | | / ___ ||
2*pi*\/ 2 \ \\/ 3 - \/ 5 // pi*\/ 2 *\/ 3 + \/ 5 *\/ 3 - \/ 5 \ \\/ 3 + \/ 5 //
------------------------------------------ + ------------------------------------------ + ------------------------------------------ + ----------------------------------------------------------------
___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5 10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5 10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5 10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5
$$\frac{\sqrt{2} \sqrt{3 - \sqrt{5}} \sqrt{\sqrt{5} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{\sqrt{5} + 3}} \right)}\right)}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}} + \frac{2 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{3 - \sqrt{5}}} \right)}\right)}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}} + \frac{2 \sqrt{2} \pi}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}} + \frac{\sqrt{2} \pi \sqrt{3 - \sqrt{5}} \sqrt{\sqrt{5} + 3}}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}}$$
=
=
/ / ___ \\ ___________ ___________ / / ___ \\
___ | |\/ 2 *tan(1/2)|| ___ / ___ / ___ | |\/ 2 *tan(1/2)||
2*\/ 2 *|-pi + atan|--------------|| \/ 2 *\/ 3 + \/ 5 *\/ 3 - \/ 5 *|-pi + atan|--------------||
| | ___________|| ___________ ___________ | | ___________||
___ | | / ___ || ___ / ___ / ___ | | / ___ ||
2*pi*\/ 2 \ \\/ 3 - \/ 5 // pi*\/ 2 *\/ 3 + \/ 5 *\/ 3 - \/ 5 \ \\/ 3 + \/ 5 //
------------------------------------------ + ------------------------------------------ + ------------------------------------------ + ----------------------------------------------------------------
___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5 10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5 10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5 10*\/ 3 - \/ 5 + 2*\/ 5 *\/ 3 - \/ 5
$$\frac{\sqrt{2} \sqrt{3 - \sqrt{5}} \sqrt{\sqrt{5} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{\sqrt{5} + 3}} \right)}\right)}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}} + \frac{2 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{\sqrt{3 - \sqrt{5}}} \right)}\right)}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}} + \frac{2 \sqrt{2} \pi}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}} + \frac{\sqrt{2} \pi \sqrt{3 - \sqrt{5}} \sqrt{\sqrt{5} + 3}}{2 \sqrt{5} \sqrt{3 - \sqrt{5}} + 10 \sqrt{3 - \sqrt{5}}}$$
2*pi*sqrt(2)/(10*sqrt(3 - sqrt(5)) + 2*sqrt(5)*sqrt(3 - sqrt(5))) + 2*sqrt(2)*(-pi + atan(sqrt(2)*tan(1/2)/sqrt(3 - sqrt(5))))/(10*sqrt(3 - sqrt(5)) + 2*sqrt(5)*sqrt(3 - sqrt(5))) + pi*sqrt(2)*sqrt(3 + sqrt(5))*sqrt(3 - sqrt(5))/(10*sqrt(3 - sqrt(5)) + 2*sqrt(5)*sqrt(3 - sqrt(5))) + sqrt(2)*sqrt(3 + sqrt(5))*sqrt(3 - sqrt(5))*(-pi + atan(sqrt(2)*tan(1/2)/sqrt(3 + sqrt(5))))/(10*sqrt(3 - sqrt(5)) + 2*sqrt(5)*sqrt(3 - sqrt(5)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.