Dickson Burton
09/08/2024 · Primary School
\( \int ( \tan \sigma \csc \theta \cos \alpha ) d \theta \)
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Step-by-step Solution
Para resolver la integral \( \int ( \tan \sigma \csc \theta \cos \alpha ) d \theta \), primero podemos notar que \( \tan \sigma \) y \( \cos \alpha \) son constantes con respecto a \( \theta \). Por lo tanto, podemos sacarlas de la integral:
\[
\tan \sigma \cos \alpha \int \csc \theta \, d\theta
\]
Ahora, necesitamos calcular la integral de \( \csc \theta \). La integral de \( \csc \theta \) es:
\[
\int \csc \theta \, d\theta = -\ln | \csc \theta + \cot \theta | + C
\]
donde \( C \) es la constante de integración. Sustituyendo esto en nuestra integral original, tenemos:
\[
\tan \sigma \cos \alpha \left( -\ln | \csc \theta + \cot \theta | + C \right)
\]
Finalmente, podemos escribir la solución de la integral como:
\[
-\tan \sigma \cos \alpha \ln | \csc \theta + \cot \theta | + C'
\]
donde \( C' = \tan \sigma \cos \alpha C \) es una nueva constante de integración.
Por lo tanto, la solución final es:
\[
-\tan \sigma \cos \alpha \ln | \csc \theta + \cot \theta | + C'
\]
Quick Answer
La solución de la integral \( \int ( \tan \sigma \csc \theta \cos \alpha ) d \theta \) es:
\[
-\tan \sigma \cos \alpha \ln | \csc \theta + \cot \theta | + C'
\]
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