Sr Examen
Integral de 1/(1+4cosx) dx
Solución
Ha introducido
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1 / | | 1 | ------------ dx | 1 + 4*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{4 \cos{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + 4*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ | \/ 15 /x\| ____ |\/ 15 /x\| | \/ 15 *log|- ------ + tan|-|| \/ 15 *log|------ + tan|-|| | 1 \ 3 \2// \ 3 \2// | ------------ dx = C - ----------------------------- + --------------------------- | 1 + 4*cos(x) 15 15 | /
$$\int \frac{1}{4 \cos{\left(x \right)} + 1}\, dx = C - \frac{\sqrt{15} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{15}}{3} \right)}}{15} + \frac{\sqrt{15} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{15}}{3} \right)}}{15}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | | \/ 15 || ____ |\/ 15 | ____ | |\/ 15 || ____ |\/ 15 |
\/ 15 *|pi*I + log|-tan(1/2) + ------|| \/ 15 *log|------| \/ 15 *|pi*I + log|------|| \/ 15 *log|------ + tan(1/2)|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- --------------------------------------- - ------------------ + --------------------------- + -----------------------------
15 15 15 15
$$- \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{15} + \frac{\sqrt{15} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{15}}{3} \right)}}{15} - \frac{\sqrt{15} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{15} + \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{15}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | | \/ 15 || ____ |\/ 15 | ____ | |\/ 15 || ____ |\/ 15 |
\/ 15 *|pi*I + log|-tan(1/2) + ------|| \/ 15 *log|------| \/ 15 *|pi*I + log|------|| \/ 15 *log|------ + tan(1/2)|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- --------------------------------------- - ------------------ + --------------------------- + -----------------------------
15 15 15 15
$$- \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{15} + \frac{\sqrt{15} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{15}}{3} \right)}}{15} - \frac{\sqrt{15} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{15} + \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{15}$$
-sqrt(15)*(pi*i + log(-tan(1/2) + sqrt(15)/3))/15 - sqrt(15)*log(sqrt(15)/3)/15 + sqrt(15)*(pi*i + log(sqrt(15)/3))/15 + sqrt(15)*log(sqrt(15)/3 + tan(1/2))/15
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.