Simplify \( \left(\frac{2}{\sqrt{x}}+10^{-2 x}\right) d x \)

To simplify \( \left(\frac{2}{\sqrt{x}} + 10^{-2x}\right) dx \), we can first split it into two separate parts. We rewrite the expression as follows: \[ \left(\frac{2}{\sqrt{x}} + 10^{-2x}\right) dx = \frac{2}{\sqrt{x}} dx + 10^{-2x} dx \] Now, we can treat each part individually. The first term, \( \frac{2}{\sqrt{x}} dx \), suggests a substitution could help with the integration process, often yielding a logarithmic result. The second term, \( 10^{-2x} dx \), would be integrated using the concept of exponential functions. To summarize the overall appearance, we have: \[ \int \left(\frac{2}{\sqrt{x}} + 10^{-2x}\right) dx = \int \frac{2}{\sqrt{x}} dx + \int 10^{-2x} dx \] This leads to: 1. The integral of \( \frac{2}{\sqrt{x}} \) gives \( 4\sqrt{x} + C_1 \). 2. The integral of \( 10^{-2x} \) can be computed using the formula for exponential integrals, giving \( -\frac{1}{2\ln(10)} 10^{-2x} + C_2 \). In conclusion, combining these results will produce the full simplified and integrated expression. So the simplified expression can be traced out clearly with the components \( 4\sqrt{x} - \frac{1}{2\ln(10)} 10^{-2x} + C \).