Integral de tanx*tan(x+a) dx
Solución
Solución detallada
Vuelva a escribir el integrando:
tan ( x ) tan ( a + x ) = tan ( a ) tan ( x ) − tan ( a ) tan ( x ) + 1 + tan 2 ( x ) − tan ( a ) tan ( x ) + 1 \tan{\left(x \right)} \tan{\left(a + x \right)} = \frac{\tan{\left(a \right)} \tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{2}{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} tan ( x ) tan ( a + x ) = − t a n ( a ) t a n ( x ) + 1 t a n ( a ) t a n ( x ) + − t a n ( a ) t a n ( x ) + 1 t a n 2 ( x )
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ tan ( a ) tan ( x ) − tan ( a ) tan ( x ) + 1 d x = tan ( a ) ∫ tan ( x ) − tan ( a ) tan ( x ) + 1 d x \int \frac{\tan{\left(a \right)} \tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1}\, dx = \tan{\left(a \right)} \int \frac{\tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1}\, dx ∫ − t a n ( a ) t a n ( x ) + 1 t a n ( a ) t a n ( x ) d x = tan ( a ) ∫ − t a n ( a ) t a n ( x ) + 1 t a n ( x ) d x
Vuelva a escribir el integrando:
tan ( x ) − tan ( a ) tan ( x ) + 1 = − tan ( x ) tan ( a ) tan ( x ) − 1 \frac{\tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} = - \frac{\tan{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1} − t a n ( a ) t a n ( x ) + 1 t a n ( x ) = − t a n ( a ) t a n ( x ) − 1 t a n ( x )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − tan ( x ) tan ( a ) tan ( x ) − 1 ) d x = − ∫ tan ( x ) tan ( a ) tan ( x ) − 1 d x \int \left(- \frac{\tan{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\, dx ∫ ( − t a n ( a ) t a n ( x ) − 1 t a n ( x ) ) d x = − ∫ t a n ( a ) t a n ( x ) − 1 t a n ( x ) d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
{ 2 x tan ( a ) 2 tan 2 ( a ) + 2 + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 2 ( a ) + 2 − log ( tan 2 ( x ) + 1 ) 2 tan 2 ( a ) + 2 for a ≠ 0 − log ( tan 2 ( x ) + 1 ) 2 otherwese \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases} ⎩ ⎨ ⎧ 2 t a n 2 ( a ) + 2 2 x t a n ( a ) + 2 t a n 2 ( a ) + 2 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) − 2 t a n 2 ( a ) + 2 l o g ( t a n 2 ( x ) + 1 ) − 2 l o g ( t a n 2 ( x ) + 1 ) for a = 0 otherwese
Por lo tanto, el resultado es: − { 2 x tan ( a ) 2 tan 2 ( a ) + 2 + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 2 ( a ) + 2 − log ( tan 2 ( x ) + 1 ) 2 tan 2 ( a ) + 2 for a ≠ 0 − log ( tan 2 ( x ) + 1 ) 2 otherwese - \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases} − ⎩ ⎨ ⎧ 2 t a n 2 ( a ) + 2 2 x t a n ( a ) + 2 t a n 2 ( a ) + 2 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) − 2 t a n 2 ( a ) + 2 l o g ( t a n 2 ( x ) + 1 ) − 2 l o g ( t a n 2 ( x ) + 1 ) for a = 0 otherwese
Por lo tanto, el resultado es: − ( { 2 x tan ( a ) 2 tan 2 ( a ) + 2 + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 2 ( a ) + 2 − log ( tan 2 ( x ) + 1 ) 2 tan 2 ( a ) + 2 for a ≠ 0 − log ( tan 2 ( x ) + 1 ) 2 otherwese ) tan ( a ) - \left(\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases}\right) \tan{\left(a \right)} − ⎩ ⎨ ⎧ 2 t a n 2 ( a ) + 2 2 x t a n ( a ) + 2 t a n 2 ( a ) + 2 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) − 2 t a n 2 ( a ) + 2 l o g ( t a n 2 ( x ) + 1 ) − 2 l o g ( t a n 2 ( x ) + 1 ) for a = 0 otherwese tan ( a )
Vuelva a escribir el integrando:
tan 2 ( x ) − tan ( a ) tan ( x ) + 1 = − tan 2 ( x ) tan ( a ) tan ( x ) − 1 \frac{\tan^{2}{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} = - \frac{\tan^{2}{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1} − t a n ( a ) t a n ( x ) + 1 t a n 2 ( x ) = − t a n ( a ) t a n ( x ) − 1 t a n 2 ( x )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − tan 2 ( x ) tan ( a ) tan ( x ) − 1 ) d x = − ∫ tan 2 ( x ) tan ( a ) tan ( x ) − 1 d x \int \left(- \frac{\tan^{2}{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan^{2}{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\, dx ∫ ( − t a n ( a ) t a n ( x ) − 1 t a n 2 ( x ) ) d x = − ∫ t a n ( a ) t a n ( x ) − 1 t a n 2 ( x ) d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
{ 2 x tan ( a ) 2 tan 3 ( a ) + 2 tan ( a ) + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 3 ( a ) + 2 tan ( a ) + log ( tan 2 ( x ) + 1 ) tan 2 ( a ) 2 tan 3 ( a ) + 2 tan ( a ) for a ≠ 0 x − tan ( x ) otherwese \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwese} \end{cases} ⎩ ⎨ ⎧ 2 t a n 3 ( a ) + 2 t a n ( a ) 2 x t a n ( a ) + 2 t a n 3 ( a ) + 2 t a n ( a ) 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) + 2 t a n 3 ( a ) + 2 t a n ( a ) l o g ( t a n 2 ( x ) + 1 ) t a n 2 ( a ) x − tan ( x ) for a = 0 otherwese
Por lo tanto, el resultado es: − { 2 x tan ( a ) 2 tan 3 ( a ) + 2 tan ( a ) + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 3 ( a ) + 2 tan ( a ) + log ( tan 2 ( x ) + 1 ) tan 2 ( a ) 2 tan 3 ( a ) + 2 tan ( a ) for a ≠ 0 x − tan ( x ) otherwese - \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwese} \end{cases} − ⎩ ⎨ ⎧ 2 t a n 3 ( a ) + 2 t a n ( a ) 2 x t a n ( a ) + 2 t a n 3 ( a ) + 2 t a n ( a ) 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) + 2 t a n 3 ( a ) + 2 t a n ( a ) l o g ( t a n 2 ( x ) + 1 ) t a n 2 ( a ) x − tan ( x ) for a = 0 otherwese
El resultado es: − ( { 2 x tan ( a ) 2 tan 2 ( a ) + 2 + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 2 ( a ) + 2 − log ( tan 2 ( x ) + 1 ) 2 tan 2 ( a ) + 2 for a ≠ 0 − log ( tan 2 ( x ) + 1 ) 2 otherwese ) tan ( a ) − { 2 x tan ( a ) 2 tan 3 ( a ) + 2 tan ( a ) + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 3 ( a ) + 2 tan ( a ) + log ( tan 2 ( x ) + 1 ) tan 2 ( a ) 2 tan 3 ( a ) + 2 tan ( a ) for a ≠ 0 x − tan ( x ) otherwese - \left(\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases}\right) \tan{\left(a \right)} - \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwese} \end{cases} − ⎩ ⎨ ⎧ 2 t a n 2 ( a ) + 2 2 x t a n ( a ) + 2 t a n 2 ( a ) + 2 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) − 2 t a n 2 ( a ) + 2 l o g ( t a n 2 ( x ) + 1 ) − 2 l o g ( t a n 2 ( x ) + 1 ) for a = 0 otherwese tan ( a ) − ⎩ ⎨ ⎧ 2 t a n 3 ( a ) + 2 t a n ( a ) 2 x t a n ( a ) + 2 t a n 3 ( a ) + 2 t a n ( a ) 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) + 2 t a n 3 ( a ) + 2 t a n ( a ) l o g ( t a n 2 ( x ) + 1 ) t a n 2 ( a ) x − tan ( x ) for a = 0 otherwese
Ahora simplificar:
{ − ( x + log ( tan ( x ) − 1 tan ( a ) ) tan ( a ) ) for a ≠ 0 − x + log ( 1 cos 2 ( x ) ) tan ( a ) 2 + tan ( x ) otherwese \begin{cases} - (x + \frac{\log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{\tan{\left(a \right)}}) & \text{for}\: a \neq 0 \\- x + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan{\left(a \right)}}{2} + \tan{\left(x \right)} & \text{otherwese} \end{cases} ⎩ ⎨ ⎧ − ( x + t a n ( a ) l o g ( t a n ( x ) − t a n ( a ) 1 ) ) − x + 2 l o g ( c o s 2 ( x ) 1 ) t a n ( a ) + tan ( x ) for a = 0 otherwese
Añadimos la constante de integración:
{ − ( x + log ( tan ( x ) − 1 tan ( a ) ) tan ( a ) ) for a ≠ 0 − x + log ( 1 cos 2 ( x ) ) tan ( a ) 2 + tan ( x ) otherwese + c o n s t a n t \begin{cases} - (x + \frac{\log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{\tan{\left(a \right)}}) & \text{for}\: a \neq 0 \\- x + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan{\left(a \right)}}{2} + \tan{\left(x \right)} & \text{otherwese} \end{cases}+ \mathrm{constant} ⎩ ⎨ ⎧ − ( x + t a n ( a ) l o g ( t a n ( x ) − t a n ( a ) 1 ) ) − x + 2 l o g ( c o s 2 ( x ) 1 ) t a n ( a ) + tan ( x ) for a = 0 otherwese + constant
Respuesta:
{ − ( x + log ( tan ( x ) − 1 tan ( a ) ) tan ( a ) ) for a ≠ 0 − x + log ( 1 cos 2 ( x ) ) tan ( a ) 2 + tan ( x ) otherwese + c o n s t a n t \begin{cases} - (x + \frac{\log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{\tan{\left(a \right)}}) & \text{for}\: a \neq 0 \\- x + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan{\left(a \right)}}{2} + \tan{\left(x \right)} & \text{otherwese} \end{cases}+ \mathrm{constant} ⎩ ⎨ ⎧ − ( x + t a n ( a ) l o g ( t a n ( x ) − t a n ( a ) 1 ) ) − x + 2 l o g ( c o s 2 ( x ) 1 ) t a n ( a ) + tan ( x ) for a = 0 otherwese + constant
Respuesta (Indefinida)
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// / 1 \ \
// / 1 \ \ || / 2 \ 2*log|- ------ + tan(x)| |
||2*log|- ------ + tan(x)| 2 / 2 \ | || log\1 + tan (x)/ \ tan(a) / 2*x*tan(a) |
/ || \ tan(a) / tan (a)*log\1 + tan (x)/ 2*x*tan(a) | ||- ---------------- + ------------------------ + ------------- for a != 0|
| ||------------------------ + ------------------------ + -------------------- for a != 0| || 2 2 2 |
| tan(x)*tan(x + a) dx = C - |< 3 3 3 | - |< 2 + 2*tan (a) 2 + 2*tan (a) 2 + 2*tan (a) |*tan(a)
| || 2*tan (a) + 2*tan(a) 2*tan (a) + 2*tan(a) 2*tan (a) + 2*tan(a) | || |
/ || | || / 2 \ |
|| x - tan(x) otherwise | || -log\1 + tan (x)/ |
\\ / || ------------------ otherwise |
\\ 2 /
∫ tan ( x ) tan ( a + x ) d x = C − ( { 2 x tan ( a ) 2 tan 2 ( a ) + 2 + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 2 ( a ) + 2 − log ( tan 2 ( x ) + 1 ) 2 tan 2 ( a ) + 2 for a ≠ 0 − log ( tan 2 ( x ) + 1 ) 2 otherwise ) tan ( a ) − { 2 x tan ( a ) 2 tan 3 ( a ) + 2 tan ( a ) + 2 log ( tan ( x ) − 1 tan ( a ) ) 2 tan 3 ( a ) + 2 tan ( a ) + log ( tan 2 ( x ) + 1 ) tan 2 ( a ) 2 tan 3 ( a ) + 2 tan ( a ) for a ≠ 0 x − tan ( x ) otherwise \int \tan{\left(x \right)} \tan{\left(a + x \right)}\, dx = C - \left(\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)} - \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwise} \end{cases} ∫ tan ( x ) tan ( a + x ) d x = C − ⎩ ⎨ ⎧ 2 t a n 2 ( a ) + 2 2 x t a n ( a ) + 2 t a n 2 ( a ) + 2 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) − 2 t a n 2 ( a ) + 2 l o g ( t a n 2 ( x ) + 1 ) − 2 l o g ( t a n 2 ( x ) + 1 ) for a = 0 otherwise tan ( a ) − ⎩ ⎨ ⎧ 2 t a n 3 ( a ) + 2 t a n ( a ) 2 x t a n ( a ) + 2 t a n 3 ( a ) + 2 t a n ( a ) 2 l o g ( t a n ( x ) − t a n ( a ) 1 ) + 2 t a n 3 ( a ) + 2 t a n ( a ) l o g ( t a n 2 ( x ) + 1 ) t a n 2 ( a ) x − tan ( x ) for a = 0 otherwise
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
