Sr Examen
Integral de 1/(А-tgx^2)(1/2) dx
Solución
Ha introducido
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1 / | | 1 | --------------- dx | / 2 \ | \a - tan (x)/*2 | / 0
$$\int\limits_{0}^{1} \frac{1}{2 \left(a - \tan^{2}{\left(x \right)}\right)}\, dx$$
Integral(1/((a - tan(x)^2)*2), (x, 0, 1))
Solución detallada
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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Ahora simplificar:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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/ 2
| x tan(x) x*tan (x)
| - ------------- - ------------- - ------------- for a = -1
| 2 2 2
| 2 + 2*tan (x) 2 + 2*tan (x) 2 + 2*tan (x)
|
| 1
< x + ------ for a = 0
| tan(x)
|
| / ___ \ / ___ \ 2 / ___ \ 2 / ___ \ / ___ \ / ___ \ ___ 5/2 3/2
| log\\/ a + tan(x)/ log\- \/ a + tan(x)/ a *log\\/ a + tan(x)/ a *log\- \/ a + tan(x)/ 2*a*log\- \/ a + tan(x)/ 2*a*log\\/ a + tan(x)/ 2*x*\/ a 2*x*a 4*x*a
/ |---------------------------------- - ---------------------------------- + ---------------------------------- - ---------------------------------- - ---------------------------------- + ---------------------------------- + ---------------------------------- + ---------------------------------- + ---------------------------------- otherwise
| | ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2
| 1 \2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a
| --------------- dx = C + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| / 2 \ 2
| \a - tan (x)/*2
|
/
$$\int \frac{1}{2 \left(a - \tan^{2}{\left(x \right)}\right)}\, dx = C + \frac{\begin{cases} - \frac{x \tan^{2}{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2} - \frac{x}{2 \tan^{2}{\left(x \right)} + 2} - \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2} & \text{for}\: a = -1 \\x + \frac{1}{\tan{\left(x \right)}} & \text{for}\: a = 0 \\\frac{2 a^{\frac{5}{2}} x}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{4 a^{\frac{3}{2}} x}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{2 \sqrt{a} x}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{a^{2} \log{\left(- \sqrt{a} + \tan{\left(x \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{a^{2} \log{\left(\sqrt{a} + \tan{\left(x \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{2 a \log{\left(- \sqrt{a} + \tan{\left(x \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{2 a \log{\left(\sqrt{a} + \tan{\left(x \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{\log{\left(- \sqrt{a} + \tan{\left(x \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{\log{\left(\sqrt{a} + \tan{\left(x \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} & \text{otherwise} \end{cases}}{2}$$
Respuesta
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/ 2 | 1 tan (1) tan(1) | - ----------------- - ----------------- - ----------------- for a = -1 | / 2 \ / 2 \ / 2 \ | 2*\2 + 2*tan (1)/ 2*\2 + 2*tan (1)/ 2*\2 + 2*tan (1)/ | < -oo for a = 0 | | ___ 5/2 / ___\ / ___ \ 3/2 / ___\ / ___ \ / ___\ / ___ \ 2 / ___\ 2 / ___ \ / ___\ / ___ \ 2 / ___\ 2 / ___ \ | \/ a a log\-\/ a / log\\/ a + tan(1)/ 2*a log\\/ a / log\- \/ a + tan(1)/ a*log\-\/ a / a*log\\/ a + tan(1)/ a *log\-\/ a / a *log\\/ a + tan(1)/ a*log\\/ a / a*log\- \/ a + tan(1)/ a *log\\/ a / a *log\- \/ a + tan(1)/ |---------------------------------- + ---------------------------------- + -------------------------------------- + -------------------------------------- + ---------------------------------- - -------------------------------------- - -------------------------------------- + ---------------------------------- + ---------------------------------- + -------------------------------------- + -------------------------------------- - ---------------------------------- - ---------------------------------- - -------------------------------------- - -------------------------------------- otherwise | ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ \2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a /
$$\begin{cases} - \frac{\tan^{2}{\left(1 \right)}}{2 \left(2 + 2 \tan^{2}{\left(1 \right)}\right)} - \frac{\tan{\left(1 \right)}}{2 \left(2 + 2 \tan^{2}{\left(1 \right)}\right)} - \frac{1}{2 \left(2 + 2 \tan^{2}{\left(1 \right)}\right)} & \text{for}\: a = -1 \\-\infty & \text{for}\: a = 0 \\\frac{a^{\frac{5}{2}}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{2 a^{\frac{3}{2}}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{\sqrt{a}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{a^{2} \log{\left(- \sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{a^{2} \log{\left(\sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{a^{2} \log{\left(- \sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} + \frac{a^{2} \log{\left(\sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} + \frac{a \log{\left(- \sqrt{a} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{a \log{\left(\sqrt{a} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{a \log{\left(- \sqrt{a} + \tan{\left(1 \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{a \log{\left(\sqrt{a} + \tan{\left(1 \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{\log{\left(- \sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{\log{\left(\sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{\log{\left(- \sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} + \frac{\log{\left(\sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} & \text{otherwise} \end{cases}$$
=
=
/ 2 | 1 tan (1) tan(1) | - ----------------- - ----------------- - ----------------- for a = -1 | / 2 \ / 2 \ / 2 \ | 2*\2 + 2*tan (1)/ 2*\2 + 2*tan (1)/ 2*\2 + 2*tan (1)/ | < -oo for a = 0 | | ___ 5/2 / ___\ / ___ \ 3/2 / ___\ / ___ \ / ___\ / ___ \ 2 / ___\ 2 / ___ \ / ___\ / ___ \ 2 / ___\ 2 / ___ \ | \/ a a log\-\/ a / log\\/ a + tan(1)/ 2*a log\\/ a / log\- \/ a + tan(1)/ a*log\-\/ a / a*log\\/ a + tan(1)/ a *log\-\/ a / a *log\\/ a + tan(1)/ a*log\\/ a / a*log\- \/ a + tan(1)/ a *log\\/ a / a *log\- \/ a + tan(1)/ |---------------------------------- + ---------------------------------- + -------------------------------------- + -------------------------------------- + ---------------------------------- - -------------------------------------- - -------------------------------------- + ---------------------------------- + ---------------------------------- + -------------------------------------- + -------------------------------------- - ---------------------------------- - ---------------------------------- - -------------------------------------- - -------------------------------------- otherwise | ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ ___ 7/2 3/2 5/2 ___ 7/2 3/2 5/2 / ___ 7/2 3/2 5/2\ / ___ 7/2 3/2 5/2\ \2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\/ a + 2*a + 6*a + 6*a 2*\/ a + 2*a + 6*a + 6*a 2*\2*\/ a + 2*a + 6*a + 6*a / 2*\2*\/ a + 2*a + 6*a + 6*a /
$$\begin{cases} - \frac{\tan^{2}{\left(1 \right)}}{2 \left(2 + 2 \tan^{2}{\left(1 \right)}\right)} - \frac{\tan{\left(1 \right)}}{2 \left(2 + 2 \tan^{2}{\left(1 \right)}\right)} - \frac{1}{2 \left(2 + 2 \tan^{2}{\left(1 \right)}\right)} & \text{for}\: a = -1 \\-\infty & \text{for}\: a = 0 \\\frac{a^{\frac{5}{2}}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{2 a^{\frac{3}{2}}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{\sqrt{a}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{a^{2} \log{\left(- \sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{a^{2} \log{\left(\sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{a^{2} \log{\left(- \sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} + \frac{a^{2} \log{\left(\sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} + \frac{a \log{\left(- \sqrt{a} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{a \log{\left(\sqrt{a} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} - \frac{a \log{\left(- \sqrt{a} + \tan{\left(1 \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{a \log{\left(\sqrt{a} + \tan{\left(1 \right)} \right)}}{2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}} + \frac{\log{\left(- \sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{\log{\left(\sqrt{a} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} - \frac{\log{\left(- \sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} + \frac{\log{\left(\sqrt{a} + \tan{\left(1 \right)} \right)}}{2 \left(2 a^{\frac{7}{2}} + 6 a^{\frac{5}{2}} + 6 a^{\frac{3}{2}} + 2 \sqrt{a}\right)} & \text{otherwise} \end{cases}$$
Piecewise((-1/(2*(2 + 2*tan(1)^2)) - tan(1)^2/(2*(2 + 2*tan(1)^2)) - tan(1)/(2*(2 + 2*tan(1)^2)), a = -1), (-oo, a = 0), (sqrt(a)/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) + a^(5/2)/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) + log(-sqrt(a))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) + log(sqrt(a) + tan(1))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) + 2*a^(3/2)/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) - log(sqrt(a))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) - log(-sqrt(a) + tan(1))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) + a*log(-sqrt(a))/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) + a*log(sqrt(a) + tan(1))/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) + a^2*log(-sqrt(a))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) + a^2*log(sqrt(a) + tan(1))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) - a*log(sqrt(a))/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) - a*log(-sqrt(a) + tan(1))/(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2)) - a^2*log(sqrt(a))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))) - a^2*log(-sqrt(a) + tan(1))/(2*(2*sqrt(a) + 2*a^(7/2) + 6*a^(3/2) + 6*a^(5/2))), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.