\( 187 \int e^{x} \sin x d x \) T \( A T E \) \[ \begin{array}{ll} u=\sin x & d u=e^{x} d x \\ d u=\cos x d x & u=e^{x} \end{array} \] \[ \int u d u=u-\int u d u \] \[ \begin{aligned} =\int e^{x} \sin x d x & =\sin \cos \left(e^{x}\right)-\int e^{x}(\cos x) d x \\ & =e^{x} \sin (x)-\int e^{x} \cos x d x \\ & =\int e^{x} \cos x d x \end{aligned} \] \[ \begin{array}{ll} u=\cos x & d u=e^{x} \\ d u=-\operatorname{sen} x & v=e^{x} \end{array} \] \[ \begin{aligned} \int e^{x} \sin x d x & =e^{x} \sin x-\left[\cos x\left(e^{x}\right)-\int e^{x}(-\operatorname{sen} x) d x\right] \\ & =e^{x} \sin x-\left[e^{x} \cos x-\int-e^{x} \operatorname{sen} x d x\right] \\ & =e^{x} \sin x-e^{x} \cos x+\left[\int-e^{x} \operatorname{sen} x d x\right] \\ -2 \int e^{x} \sin x d x & =e^{x} \sin x-e^{x} \cos x+c \\ & =-\frac{1}{2} e^{x}[\sin x-\cos x]+c \end{aligned} \]

Did you know that the integral of \( e^x \sin x \) actually appears in various fields, including engineering and physics? This integral helps solve differential equations that model systems like harmonic oscillators or electrical circuits with damping. When you encounter these integrals in real-world scenarios, they may represent oscillatory motion or waveforms, showcasing the beauty of mathematics in explaining natural phenomena! When integrating functions like \( e^x \sin x \), it's easy to get tangled in endless integration by parts. A common mistake is forgetting to rearrange terms correctly or misapplying integration limits. Remember, when you integrate by parts, be sure to track each function and its differential accurately. A little oversight can lead to a big tangle, so double-check your work and keep consistent notation throughout the process!