Sr Examen
Integral de 1/(4*sinx+7*cosx+2) dx
Solución
Ha introducido
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1 / | | 1 | ----------------------- dx | 4*sin(x) + 7*cos(x) + 2 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}\right) + 2}\, dx$$
Integral(1/(4*sin(x) + 7*cos(x) + 2), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ | 4 \/ 61 /x\| ____ | 4 \/ 61 /x\| | \/ 61 *log|- - - ------ + tan|-|| \/ 61 *log|- - + ------ + tan|-|| | 1 \ 5 5 \2// \ 5 5 \2// | ----------------------- dx = C - --------------------------------- + --------------------------------- | 4*sin(x) + 7*cos(x) + 2 61 61 | /
$$\int \frac{1}{\left(4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}\right) + 2}\, dx = C + \frac{\sqrt{61} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{61}}{5} - \frac{4}{5} \right)}}{61}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |4 \/ 61 || ____ | 4 \/ 61 | ____ | |4 \/ 61 || ____ | 4 \/ 61 |
\/ 61 *|pi*I + log|- - tan(1/2) + ------|| \/ 61 *log|- - + ------| \/ 61 *|pi*I + log|- + ------|| \/ 61 *log|- - + ------ + tan(1/2)|
\ \5 5 // \ 5 5 / \ \5 5 // \ 5 5 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
61 61 61 61
$$\frac{\sqrt{61} \log{\left(- \frac{4}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \log{\left(- \frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{4}{5} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61} + \frac{\sqrt{61} \left(\log{\left(\frac{4}{5} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |4 \/ 61 || ____ | 4 \/ 61 | ____ | |4 \/ 61 || ____ | 4 \/ 61 |
\/ 61 *|pi*I + log|- - tan(1/2) + ------|| \/ 61 *log|- - + ------| \/ 61 *|pi*I + log|- + ------|| \/ 61 *log|- - + ------ + tan(1/2)|
\ \5 5 // \ 5 5 / \ \5 5 // \ 5 5 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
61 61 61 61
$$\frac{\sqrt{61} \log{\left(- \frac{4}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \log{\left(- \frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{4}{5} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61} + \frac{\sqrt{61} \left(\log{\left(\frac{4}{5} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61}$$
-sqrt(61)*(pi*i + log(4/5 - tan(1/2) + sqrt(61)/5))/61 - sqrt(61)*log(-4/5 + sqrt(61)/5)/61 + sqrt(61)*(pi*i + log(4/5 + sqrt(61)/5))/61 + sqrt(61)*log(-4/5 + sqrt(61)/5 + tan(1/2))/61
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.