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