Sr Examen
Integral de 1/(4+sinxcosx) dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------------- dx | 4 + sin(x)*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(x \right)} \cos{\left(x \right)} + 4}\, dx$$
Integral(1/(4 + sin(x)*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ / ___ /x\\\ / /x pi\ / ___ /x\\\
| |- - --| | ___ 2*\/ 7 *tan|-||| | |- - --| | ___ 4*\/ 7 *tan|-|||
/ ___ | |2 2 | | \/ 7 \2/|| ___ | |2 2 | |\/ 7 \2/||
| 2*\/ 7 *|pi*floor|------| + atan|- ----- + --------------|| 2*\/ 7 *|pi*floor|------| + atan|----- + --------------||
| 1 \ \ pi / \ 7 7 // \ \ pi / \ 7 7 //
| ----------------- dx = C + ----------------------------------------------------------- + ---------------------------------------------------------
| 4 + sin(x)*cos(x) 21 21
|
/
$$\int \frac{1}{\sin{\left(x \right)} \cos{\left(x \right)} + 4}\, dx = C + \frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\frac{2 \sqrt{7} \tan{\left(\frac{x}{2} \right)}}{7} - \frac{\sqrt{7}}{7} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{21} + \frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\frac{4 \sqrt{7} \tan{\left(\frac{x}{2} \right)}}{7} + \frac{\sqrt{7}}{7} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{21}$$
Gráfica
Respuesta
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/ / ___\\ / / ___\\ / / ___ ___ \\ / / ___ ___ \\
___ | |\/ 7 || ___ | |\/ 7 || ___ | |\/ 7 2*\/ 7 *tan(1/2)|| ___ | |\/ 7 4*\/ 7 *tan(1/2)||
2*\/ 7 *|-pi - atan|-----|| 2*\/ 7 *|-pi + atan|-----|| 2*\/ 7 *|-pi - atan|----- - ----------------|| 2*\/ 7 *|-pi + atan|----- + ----------------||
\ \ 7 // \ \ 7 // \ \ 7 7 // \ \ 7 7 //
- --------------------------- - --------------------------- + ---------------------------------------------- + ----------------------------------------------
21 21 21 21
$$\frac{2 \sqrt{7} \left(- \pi - \operatorname{atan}{\left(- \frac{2 \sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} + \frac{\sqrt{7}}{7} \right)}\right)}{21} + \frac{2 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7}}{7} + \frac{4 \sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} \right)}\right)}{21} - \frac{2 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}\right)}{21} - \frac{2 \sqrt{7} \left(- \pi - \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}\right)}{21}$$
=
=
/ / ___\\ / / ___\\ / / ___ ___ \\ / / ___ ___ \\
___ | |\/ 7 || ___ | |\/ 7 || ___ | |\/ 7 2*\/ 7 *tan(1/2)|| ___ | |\/ 7 4*\/ 7 *tan(1/2)||
2*\/ 7 *|-pi - atan|-----|| 2*\/ 7 *|-pi + atan|-----|| 2*\/ 7 *|-pi - atan|----- - ----------------|| 2*\/ 7 *|-pi + atan|----- + ----------------||
\ \ 7 // \ \ 7 // \ \ 7 7 // \ \ 7 7 //
- --------------------------- - --------------------------- + ---------------------------------------------- + ----------------------------------------------
21 21 21 21
$$\frac{2 \sqrt{7} \left(- \pi - \operatorname{atan}{\left(- \frac{2 \sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} + \frac{\sqrt{7}}{7} \right)}\right)}{21} + \frac{2 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7}}{7} + \frac{4 \sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} \right)}\right)}{21} - \frac{2 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}\right)}{21} - \frac{2 \sqrt{7} \left(- \pi - \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}\right)}{21}$$
-2*sqrt(7)*(-pi - atan(sqrt(7)/7))/21 - 2*sqrt(7)*(-pi + atan(sqrt(7)/7))/21 + 2*sqrt(7)*(-pi - atan(sqrt(7)/7 - 2*sqrt(7)*tan(1/2)/7))/21 + 2*sqrt(7)*(-pi + atan(sqrt(7)/7 + 4*sqrt(7)*tan(1/2)/7))/21
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.