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