Question
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Answer
La integral es \( -\cos(\sec(x)) + C \).
Solution
Sea
\[
u = \sec(x)
\]
entonces,
\[
\frac{du}{dx} = \sec(x)\tan(x) \quad \Rightarrow \quad du = \sec(x)\tan(x)\,dx.
\]
La integral se transforma en
\[
\int \sec(x)\tan(x)\sin(\sec(x))\,dx = \int \sin(u)\,du.
\]
La integral de \(\sin(u)\) es
\[
\int \sin(u)\,du = -\cos(u) + C.
\]
Volviendo a la variable original, tenemos
\[
-\cos(u) + C = -\cos(\sec(x)) + C.
\]
Respuesta final:
\[
-\cos(\sec(x)) + C.
\]
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Simplify this solution Extra Insights
To tackle the integral \( \int \sec ( x ) \tan ( x ) \sin ( \sec ( x ) ) d x \), let’s consider a simple substitution that might help simplify things. Notably, since secant and tangent derivatives are connected, you can set \( u = \sec(x) \). Then \( du = \sec(x) \tan(x) \, dx \), making the integral easier to handle as: \[ \int \sin(u) \, du \] Now you can focus on the integral \( -\cos(u) + C \), which in terms of \( x \) translates back to: \[ -\cos(\sec(x)) + C \] This will help you evaluate and express the original integral simply! Remember to always consider potential substitutions that align with derivatives you recognize.
