Sr Examen
Integral de pi*1/(1+9x^2)*1/arctg^-2(3x) dx
Solución
Ha introducido
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1 / | | / pi \ | |--------| | | 2| | \1 + 9*x / | ------------ dx | / 1 \ | |----------| | | 2 | | \atan (3*x)/ | / 1/3
$$\int\limits_{\frac{1}{3}}^{1} \frac{\pi \frac{1}{9 x^{2} + 1}}{\frac{1}{\operatorname{atan}^{2}{\left(3 x \right)}}}\, dx$$
Integral((pi/(1 + 9*x^2))/atan(3*x)^(-2), (x, 1/3, 1))
Solución detallada
-
Añadimos la constante de integración:
TrigSubstitutionRule(theta=_theta, func=tan(_theta)/3, rewritten=pi*atan(tan(_theta))**2/3, substep=ConstantTimesRule(constant=pi/3, other=atan(tan(_theta))**2, substep=URule(u_var=_u, u_func=atan(tan(_theta)), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=atan(tan(_theta))**2, symbol=_theta), context=pi*atan(tan(_theta))**2/3, symbol=_theta), restriction=True, context=(pi/(9*x**2 + 1))/atan(3*x)**(-2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/ | | / pi \ | |--------| | | 2| 3 | \1 + 9*x / pi*atan (3*x) | ------------ dx = C + ------------- | / 1 \ 9 | |----------| | | 2 | | \atan (3*x)/ | /
$$\int \frac{\pi \frac{1}{9 x^{2} + 1}}{\frac{1}{\operatorname{atan}^{2}{\left(3 x \right)}}}\, dx = C + \frac{\pi \operatorname{atan}^{3}{\left(3 x \right)}}{9}$$
Gráfica
Respuesta
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4 3 pi pi*atan (3) - --- + ----------- 576 9
$$- \frac{\pi^{4}}{576} + \frac{\pi \operatorname{atan}^{3}{\left(3 \right)}}{9}$$
=
=
4 3 pi pi*atan (3) - --- + ----------- 576 9
$$- \frac{\pi^{4}}{576} + \frac{\pi \operatorname{atan}^{3}{\left(3 \right)}}{9}$$
-pi^4/576 + pi*atan(3)^3/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.