Sr Examen
Integral de 1/((sin(x)^2+8*sin(x)*cos(x)+12*(cos(x))^2)) dx
Solución
Ha introducido
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1 / | | 1 | -------------------------------------- dx | 2 2 | sin (x) + 8*sin(x)*cos(x) + 12*cos (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(\sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) + 12 \cos^{2}{\left(x \right)}}\, dx$$
Integral(1/(sin(x)^2 + (8*sin(x))*cos(x) + 12*cos(x)^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ / /x\ 2/x\\ / 2/x\ /x\\ | log|-3 - tan|-| + 3*tan |-|| log|-1 + tan |-| - tan|-|| | 1 \ \2/ \2// \ \2/ \2// | -------------------------------------- dx = C - ---------------------------- + -------------------------- | 2 2 4 4 | sin (x) + 8*sin(x)*cos(x) + 12*cos (x) | /
$$\int \frac{1}{\left(\sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) + 12 \cos^{2}{\left(x \right)}}\, dx = C + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\log{\left(3 \tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} - 3 \right)}}{4}$$
Gráfica
Respuesta
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/ 2 \ / 2 \
log\3 - 3*tan (1/2) + tan(1/2)/ log(3) log\1 - tan (1/2) + tan(1/2)/
- ------------------------------- + ------ + -----------------------------
4 4 4
$$- \frac{\log{\left(- 3 \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)}}{4} + \frac{\log{\left(3 \right)}}{4}$$
=
=
/ 2 \ / 2 \
log\3 - 3*tan (1/2) + tan(1/2)/ log(3) log\1 - tan (1/2) + tan(1/2)/
- ------------------------------- + ------ + -----------------------------
4 4 4
$$- \frac{\log{\left(- 3 \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)}}{4} + \frac{\log{\left(3 \right)}}{4}$$
-log(3 - 3*tan(1/2)^2 + tan(1/2))/4 + log(3)/4 + log(1 - tan(1/2)^2 + tan(1/2))/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.