Evalúa las siguientes integrales utilizando la integración 2. \( \int \ln 5 x d x \). 3. \( \int x \sec ^{2} x d x \). 4. \( \int x 3^{x} d x \). 5. \( \int \frac{\operatorname{sen} 2 x}{e^{x}} \).

Calculate the integral \( \int \ln(5x) dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int \ln{\left(5x\right)} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=\ln{\left(5x\right)}\\&dv=dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=\frac{1}{x}dx\\&dv=dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=\frac{1}{x}dx\\&v=x\end{align}\) - step4: Substitute the values into formula: \(\ln{\left(5x\right)}\times x-\int \frac{1}{x}\times x dx\) - step5: Calculate: \(x\ln{\left(5x\right)}-\int 1 dx\) - step6: Evaluate the integral: \(x\ln{\left(5x\right)}-x\) - step7: Add the constant of integral C: \(x\ln{\left(5x\right)}-x + C, C \in \mathbb{R}\) Calculate the integral \( \int x \sec^2(x) dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x\sec^{2}\left(x\right) dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=\sec^{2}\left(x\right)dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=dx\\&dv=\sec^{2}\left(x\right)dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=dx\\&v=\tan\left(x\right)\end{align}\) - step4: Substitute the values into formula: \(x\tan\left(x\right)-\int 1\times \tan\left(x\right) dx\) - step5: Calculate: \(x\tan\left(x\right)-\int \tan\left(x\right) dx\) - step6: Evaluate the integral: \(x\tan\left(x\right)-\ln{\left(\left|\sec\left(x\right)\right|\right)}\) - step7: Add the constant of integral C: \(x\tan\left(x\right)-\ln{\left(\left|\sec\left(x\right)\right|\right)} + C, C \in \mathbb{R}\) Calculate the integral \( \int x 3^x dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x\times 3^{x} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=3^{x}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=dx\\&dv=3^{x}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=dx\\&v=\frac{3^{x}}{\ln{\left(3\right)}}\end{align}\) - step4: Substitute the values into formula: \(x\times \frac{3^{x}}{\ln{\left(3\right)}}-\int 1\times \frac{3^{x}}{\ln{\left(3\right)}} dx\) - step5: Calculate: \(\frac{x\times 3^{x}}{\ln{\left(3\right)}}-\int \frac{3^{x}}{\ln{\left(3\right)}} dx\) - step6: Evaluate the integral: \(\frac{x\times 3^{x}}{\ln{\left(3\right)}}-\frac{3^{x}}{\left(\ln{\left(3\right)}\right)^{2}}\) - step7: Simplify the expression: \(\frac{3^{x}x}{\ln{\left(3\right)}}-\frac{3^{x}}{\left(\ln{\left(3\right)}\right)^{2}}\) - step8: Simplify: \(\frac{3^{x}x}{\ln{\left(3\right)}}-\frac{3^{x}}{1.206949}\) - step9: Add the constant of integral C: \(\frac{3^{x}x}{\ln{\left(3\right)}}-\frac{3^{x}}{1.206949} + C, C \in \mathbb{R}\) Calculate the integral \( \int \frac{\sin(2x)}{e^x} dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{\sin\left(2x\right)}{e^{x}} dx\) - step1: Simplify the expression: \(\int \sin\left(2x\right)\times e^{-x} dx\) - step2: Evaluate the integral: \(\frac{e^{-x}}{\left(-1\right)^{2}+2^{2}}\times \left(-\sin\left(2x\right)-2\cos\left(2x\right)\right)\) - step3: Evaluate the power: \(\frac{e^{-x}}{1+2^{2}}\times \left(-\sin\left(2x\right)-2\cos\left(2x\right)\right)\) - step4: Add the numbers: \(\frac{e^{-x}}{5}\left(-\sin\left(2x\right)-2\cos\left(2x\right)\right)\) - step5: Simplify the expression: \(\frac{1}{5e^{x}}\times \left(-\sin\left(2x\right)-2\cos\left(2x\right)\right)\) - step6: Rewrite the expression: \(\frac{1}{5e^{x}}\times \left(-\left(\sin\left(2x\right)+2\cos\left(2x\right)\right)\right)\) - step7: Multiply the terms: \(-\frac{1}{5e^{x}}\times \left(\sin\left(2x\right)+2\cos\left(2x\right)\right)\) - step8: Multiply the terms: \(-\frac{\sin\left(2x\right)+2\cos\left(2x\right)}{5e^{x}}\) - step9: Add the constant of integral C: \(-\frac{\sin\left(2x\right)+2\cos\left(2x\right)}{5e^{x}} + C, C \in \mathbb{R}\) Las integrales evaluadas son: 1. \( \int \ln(5x) dx = x\ln(5x) - x + C \) 2. \( \int x \sec^2(x) dx = x\tan(x) - \ln(|\sec(x)|) + C \) 3. \( \int x 3^x dx = \frac{3^x x}{\ln(3)} - \frac{3^x}{1.206949} + C \) 4. \( \int \frac{\sin(2x)}{e^x} dx = -\frac{\sin(2x) + 2\cos(2x)}{5e^x} + C \)