\( = \int ( e ^ { \sin x } \cdot \frac { \sin x } { \sec x } ) d x \)
Solución de inteligencia artificial de Upstudy
Responder
Solución
Respondido por UpStudy AI y revisado por un tutor profesional


Beyond the Answer
To tackle this integral, let’s first simplify the integrand. We can rewrite \(\sec x\) as \(\frac{1}{\cos x}\), which means \(\frac{\sin x}{\sec x} = \sin x \cdot \cos x\). Thus, the integral transforms into: \[ \int e^{\sin x} \cdot (\sin x \cdot \cos x) \, dx \] This expression can further be simplified with a substitution. You can use \(u = \sin x\), where \(du = \cos x \, dx\). So, the integral becomes: \[ \int e^u \cdot u \, du \] This is now a standard integral that can be solved using integration by parts. Integration by parts states that \(\int u \, dv = uv - \int v \, du\). Here, let \(u = u\) and \(dv = e^u du\). Thus, \(du = du\) and \(v = e^u\). Applying integration by parts gives you: \[ \int u e^u \, du = u e^u - \int e^u \, du \] This results in: \[ u e^u - e^u + C \] Finally, substitute back \(u = \sin x\) to obtain the final result: \[ \sin x \cdot e^{\sin x} - e^{\sin x} + C \] There you go! The integral simplifies beautifully and you’ve now got a nice result to show off.
