Sr Examen
Integral de 1/(4cosx+3sinx-1) dx
Solución
Ha introducido
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1 / | | 1 | ----------------------- dx | 4*cos(x) + 3*sin(x) - 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 1}\, dx$$
Integral(1/(4*cos(x) + 3*sin(x) - 1), (x, 0, 1))
Respuesta (Indefinida)
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/ ___ \ / ___ \ / ___ | 3 2*\/ 6 /x\| ___ | 3 2*\/ 6 /x\| | \/ 6 *log|- - - ------- + tan|-|| \/ 6 *log|- - + ------- + tan|-|| | 1 \ 5 5 \2// \ 5 5 \2// | ----------------------- dx = C - --------------------------------- + --------------------------------- | 4*cos(x) + 3*sin(x) - 1 12 12 | /
$$\int \frac{1}{\left(3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 1}\, dx = C + \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}}{12} - \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2 \sqrt{6}}{5} - \frac{3}{5} \right)}}{12}$$
Gráfica
Respuesta
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/ / ___\\ / ___\ / / ___\\ / ___ \
___ | |3 2*\/ 6 || ___ | 3 2*\/ 6 | ___ | |3 2*\/ 6 || ___ | 3 2*\/ 6 |
\/ 6 *|pi*I + log|- - tan(1/2) + -------|| \/ 6 *log|- - + -------| \/ 6 *|pi*I + log|- + -------|| \/ 6 *log|- - + ------- + tan(1/2)|
\ \5 5 // \ 5 5 / \ \5 5 // \ 5 5 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
12 12 12 12
$$\frac{\sqrt{6} \log{\left(- \frac{3}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{2 \sqrt{6}}{5} \right)}}{12} - \frac{\sqrt{6} \log{\left(- \frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)} + i \pi\right)}{12}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___ \
___ | |3 2*\/ 6 || ___ | 3 2*\/ 6 | ___ | |3 2*\/ 6 || ___ | 3 2*\/ 6 |
\/ 6 *|pi*I + log|- - tan(1/2) + -------|| \/ 6 *log|- - + -------| \/ 6 *|pi*I + log|- + -------|| \/ 6 *log|- - + ------- + tan(1/2)|
\ \5 5 // \ 5 5 / \ \5 5 // \ 5 5 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
12 12 12 12
$$\frac{\sqrt{6} \log{\left(- \frac{3}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{2 \sqrt{6}}{5} \right)}}{12} - \frac{\sqrt{6} \log{\left(- \frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)} + i \pi\right)}{12}$$
-sqrt(6)*(pi*i + log(3/5 - tan(1/2) + 2*sqrt(6)/5))/12 - sqrt(6)*log(-3/5 + 2*sqrt(6)/5)/12 + sqrt(6)*(pi*i + log(3/5 + 2*sqrt(6)/5))/12 + sqrt(6)*log(-3/5 + 2*sqrt(6)/5 + tan(1/2))/12
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.