1. Resuelve las siguientes integrales. a. \( \int(2 x+\sqrt{x}) d x \) b. \( \int \frac{2 x}{3 x^{2}-1} d x \) c. \( \int \mathrm{e}^{-\mathrm{x}} \cdot\left(1+e^{-x}\right)^{5} d x \) d. \( \int \frac{\operatorname{sen} x \cos x}{1+\operatorname{sen}} d x \) e. \( \int x^{2} \sqrt{x}+1 \) f. f. \( \int \operatorname{sen} 2 x \cdot e^{2 x} d x \) g. \( \int \frac{3 x-1}{x^{2}-5 x} d x \)

Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{2x}{3x^{2}-1} dx\) - step1: Rewrite the expression: \(\int 2\times \frac{x}{3x^{2}-1} dx\) - step2: Use properties of integrals: \(2\times \int \frac{x}{3x^{2}-1} dx\) - step3: Use the substitution \(dx=\frac{1}{6x} dt\) to transform the integral\(:\) \(2\times \int \frac{x}{3x^{2}-1}\times \frac{1}{6x} dt\) - step4: Simplify: \(2\times \int \frac{1}{6\times 3x^{2}-6} dt\) - step5: Use the substitution \(t=3x^{2}\) to transform the integral\(:\) \(2\times \int \frac{1}{6t-6} dt\) - step6: Rewrite the expression: \(2\times \int \frac{1}{6}\times \frac{1}{t-1} dt\) - step7: Use properties of integrals: \(2\times \frac{1}{6}\times \int \frac{1}{t-1} dt\) - step8: Multiply the numbers: \(\frac{1}{3}\times \int \frac{1}{t-1} dt\) - step9: Evaluate the integral: \(\frac{1}{3}\ln{\left(\left|t-1\right|\right)}\) - step10: Substitute back: \(\frac{1}{3}\ln{\left(\left|3x^{2}-1\right|\right)}\) - step11: Add the constant of integral C: \(\frac{1}{3}\ln{\left(\left|3x^{2}-1\right|\right)} + C, C \in \mathbb{R}\) Calculate the integral \( \int(2 x+\sqrt{x}) d x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(2x+\sqrt{x}\right) dx\) - step1: Use properties of integrals: \(\int 2x dx+\int \sqrt{x} dx\) - step2: Evaluate the integral: \(x^{2}+\int \sqrt{x} dx\) - step3: Evaluate the integral: \(x^{2}+\frac{2}{3}x^{\frac{3}{2}}\) - step4: Simplify: \(x^{2}+\frac{2}{3}x\sqrt{x}\) - step5: Add the constant of integral C: \(x^{2}+\frac{2}{3}x\sqrt{x} + C, C \in \mathbb{R}\) Calculate the integral \( \int x^{2} \sqrt{x}+1 d x \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int \left(x^{2}\sqrt{x}+1\right) dx\) - step1: Use properties of integrals: \(\int x^{2}\sqrt{x} dx+\int 1 dx\) - step2: Evaluate the integral: \(\frac{2}{3}x^{\frac{7}{2}}-\frac{8}{21}x^{\frac{7}{2}}+\int 1 dx\) - step3: Evaluate the integral: \(\frac{2}{3}x^{\frac{7}{2}}-\frac{8}{21}x^{\frac{7}{2}}+x\) - step4: Calculate: \(\frac{2}{7}x^{\frac{7}{2}}+x\) - step5: Simplify: \(\frac{2}{7}x^{3}\sqrt{x}+x\) - step6: Add the constant of integral C: \(\frac{2}{7}x^{3}\sqrt{x}+x + C, C \in \mathbb{R}\) Calculate the integral \( \int \sin(2 x) \cdot e^{2 x} d x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \sin\left(2x\right)\times e^{2x} dx\) - step1: Evaluate the integral: \(\frac{e^{2x}}{2^{2}+2^{2}}\times \left(2\sin\left(2x\right)-2\cos\left(2x\right)\right)\) - step2: Add the numbers: \(\frac{e^{2x}}{8}\left(2\sin\left(2x\right)-2\cos\left(2x\right)\right)\) - step3: Rewrite the expression: \(\frac{e^{2x}}{8}\times 2\left(\sin\left(2x\right)-\cos\left(2x\right)\right)\) - step4: Reduce the fraction: \(\frac{e^{2x}}{4}\left(\sin\left(2x\right)-\cos\left(2x\right)\right)\) - step5: Multiply the terms: \(\frac{e^{2x}\left(\sin\left(2x\right)-\cos\left(2x\right)\right)}{4}\) - step6: Simplify the expression: \(\frac{e^{2x}\sin\left(2x\right)-e^{2x}\cos\left(2x\right)}{4}\) - step7: Calculate: \(\frac{e^{2x}\sin\left(2x\right)}{4}-\frac{e^{2x}\cos\left(2x\right)}{4}\) - step8: Add the constant of integral C: \(\frac{e^{2x}\sin\left(2x\right)}{4}-\frac{e^{2x}\cos\left(2x\right)}{4} + C, C \in \mathbb{R}\) Calculate the integral \( \int \frac{3 x-1}{x^{2}-5 x} d x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{3x-1}{x^{2}-5x} dx\) - step1: Rewrite the fraction: \(\int \left(\frac{1}{5x}+\frac{14}{5x-25}\right) dx\) - step2: Use properties of integrals: \(\int \frac{1}{5x} dx+\int \frac{14}{5x-25} dx\) - step3: Evaluate the integral: \(\frac{1}{5}\ln{\left(\left|x\right|\right)}+\int \frac{14}{5x-25} dx\) - step4: Evaluate the integral: \(\frac{1}{5}\ln{\left(\left|x\right|\right)}+\frac{14}{5}\ln{\left(\left|x-5\right|\right)}\) - step5: Use the logarithm base change rule: \(\ln{\left(\left|x\right|^{\frac{1}{5}}\right)}+\ln{\left(\left(x-5\right)^{\frac{14}{5}}\right)}\) - step6: Transform the expression: \(\ln{\left(\left|x\right|^{\frac{1}{5}}\times \left(x-5\right)^{\frac{14}{5}}\right)}\) - step7: Simplify: \(\ln{\left(\left|x^{\frac{1}{5}}\right|\times \left(x-5\right)^{\frac{14}{5}}\right)}\) - step8: Add the constant of integral C: \(\ln{\left(\left|x^{\frac{1}{5}}\right|\times \left(x-5\right)^{\frac{14}{5}}\right)} + C, C \in \mathbb{R}\) Aquí están las soluciones a las integrales solicitadas: a. Para la integral \( \int(2 x+\sqrt{x}) d x \): \[ \int(2 x+\sqrt{x}) d x = x^{2}+\frac{2}{3}x\sqrt{x} + C, \quad C \in \mathbb{R} \] b. Para la integral \( \int \frac{2 x}{3 x^{2}-1} d x \): \[ \int \frac{2 x}{3 x^{2}-1} d x = \frac{1}{3}\ln{(3x^{2}-1)} + C, \quad C \in \mathbb{R} \] c. La integral \( \int e^{-x} \cdot(1+e^{-x})^{5} d x \) no se pudo calcular debido a un tiempo de espera en el cálculo. d. La integral \( \int \frac{\sin x \cos x}{1+\sin x} d x \) no se pudo calcular debido a que la operación no es soportada. e. Para la integral \( \int x^{2} \sqrt{x}+1 d x \): \[ \int x^{2} \sqrt{x}+1 d x = \frac{2}{7}x^{3}\sqrt{x}+x + C, \quad C \in \mathbb{R} \] f. Para la integral \( \int \sin(2 x) \cdot e^{2 x} d x \): \[ \int \sin(2 x) \cdot e^{2 x} d x = \frac{e^{2x}\sin(2x)}{4}-\frac{e^{2x}\cos(2x)}{4} + C, \quad C \in \mathbb{R} \] g. Para la integral \( \int \frac{3 x-1}{x^{2}-5 x} d x \): \[ \int \frac{3 x-1}{x^{2}-5 x} d x = \ln{(x^{\frac{1}{5}}(x-5)^{\frac{14}{5}})} + C, \quad C \in \mathbb{R} \] Si necesitas más ayuda con la integral que no se pudo calcular o cualquier otra consulta, házmelo saber.