\( \begin{array}{ll}\text { (1) } \int\left(x^{2}+3 x-2\right) d x & \text { (4) } \int \frac{d x}{(3 x+5)^{4}} \\ \text { (5) } \int \frac{d x}{(x+1)^{3}} & \text { (5) } \int x e^{-2 x} d x \\ \text { (3) } \int x(3 x-5)^{4} d x & \text { (6) } \int x^{2} e^{x} d x \\ \text { (7) } \int x^{2} \log x d x & \text { (8) } \int 4^{x} e^{2 x} d x\end{array} \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(x^{2}+3x-2\right) dx\) - step1: Use properties of integrals: \(\int x^{2} dx+\int 3x dx+\int -2 dx\) - step2: Evaluate the integral: \(\frac{x^{3}}{3}+\int 3x dx+\int -2 dx\) - step3: Evaluate the integral: \(\frac{x^{3}}{3}+\frac{3x^{2}}{2}+\int -2 dx\) - step4: Evaluate the integral: \(\frac{x^{3}}{3}+\frac{3x^{2}}{2}-2x\) - step5: Add the constant of integral C: \(\frac{x^{3}}{3}+\frac{3x^{2}}{2}-2x + C, C \in \mathbb{R}\) Calculate the integral \( \frac{1}{(3 x+5)^{4}} \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{1}{\left(3x+5\right)^{4}} dx\) - step1: Use the substitution \(dx=\frac{1}{3} dt\) to transform the integral\(:\) \(\int \frac{1}{\left(3x+5\right)^{4}}\times \frac{1}{3} dt\) - step2: Simplify: \(\int \frac{1}{3\left(3x+5\right)^{4}} dt\) - step3: Use the substitution \(t=3x+5\) to transform the integral\(:\) \(\int \frac{1}{3t^{4}} dt\) - step4: Rewrite the expression: \(\int \frac{1}{3}\times \frac{1}{t^{4}} dt\) - step5: Use properties of integrals: \(\frac{1}{3}\times \int \frac{1}{t^{4}} dt\) - step6: Evaluate the integral: \(\frac{1}{3}\times \frac{t^{-4+1}}{-4+1}\) - step7: Simplify: \(\frac{1}{3}\times \frac{t^{-3}}{-3}\) - step8: Multiply the terms: \(-\frac{1}{3}\times \frac{t^{-3}}{3}\) - step9: Multiply the terms: \(-\frac{t^{-3}}{3\times 3}\) - step10: Multiply the terms: \(-\frac{t^{-3}}{9}\) - step11: Simplify: \(-\frac{1}{9t^{3}}\) - step12: Substitute back: \(-\frac{1}{9\left(3x+5\right)^{3}}\) - step13: Add the constant of integral C: \(-\frac{1}{9\left(3x+5\right)^{3}} + C, C \in \mathbb{R}\) Calculate the integral \( \frac{1}{(x+1)^{3}} \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{1}{\left(x+1\right)^{3}} dx\) - step1: Use the substitution \(dx=1 dt\) to transform the integral\(:\) \(\int \frac{1}{\left(x+1\right)^{3}}\times 1 dt\) - step2: Simplify: \(\int \frac{1}{\left(x+1\right)^{3}} dt\) - step3: Use the substitution \(t=x+1\) to transform the integral\(:\) \(\int \frac{1}{t^{3}} dt\) - step4: Evaluate the integral: \(\frac{t^{-3+1}}{-3+1}\) - step5: Add the numbers: \(\frac{t^{-2}}{-3+1}\) - step6: Add the numbers: \(\frac{t^{-2}}{-2}\) - step7: Substitute back: \(\frac{\left(x+1\right)^{-2}}{-2}\) - step8: Simplify: \(\frac{\frac{1}{\left(x+1\right)^{2}}}{-2}\) - step9: Add the constant of integral C: \(\frac{\frac{1}{\left(x+1\right)^{2}}}{-2} + C, C \in \mathbb{R}\) Calculate the integral \( x^{2} e^{x} \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x^{2}e^{x} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x^{2}\\&dv=e^{x}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=2xdx\\&dv=e^{x}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=2xdx\\&v=e^{x}\end{align}\) - step4: Substitute the values into formula: \(x^{2}e^{x}-\int 2xe^{x} dx\) - step5: Evaluate the integral: \(x^{2}e^{x}-2xe^{x}+2e^{x}\) - step6: Add the constant of integral C: \(x^{2}e^{x}-2xe^{x}+2e^{x} + C, C \in \mathbb{R}\) Calculate the integral \( 4^{x} e^{2 x} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int 4^{x}e^{2x} dx\) - step1: Multiply the terms: \(\int \left(4e^{2}\right)^{x} dx\) - step2: Evaluate the integral: \(\frac{\left(4e^{2}\right)^{x}}{\ln{\left(4e^{2}\right)}}\) - step3: Simplify: \(\frac{\left(4e^{2}\right)^{x}}{2\ln{\left(2e\right)}}\) - step4: Add the constant of integral C: \(\frac{\left(4e^{2}\right)^{x}}{2\ln{\left(2e\right)}} + C, C \in \mathbb{R}\) Calculate the integral \( x e^{-2 x} \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int xe^{-2x} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=e^{-2x}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=dx\\&dv=e^{-2x}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=dx\\&v=-\frac{1}{2e^{2x}}\end{align}\) - step4: Substitute the values into formula: \(x\left(-\frac{1}{2e^{2x}}\right)-\int 1\times \left(-\frac{1}{2e^{2x}}\right) dx\) - step5: Calculate: \(-\frac{x}{2e^{2x}}-\int -\frac{1}{2e^{2x}} dx\) - step6: Use properties of integrals: \(-\frac{x}{2e^{2x}}+\int \frac{1}{2e^{2x}} dx\) - step7: Evaluate the integral: \(-\frac{x}{2e^{2x}}-\frac{1}{4e^{2x}}\) - step8: Add the constant of integral C: \(-\frac{x}{2e^{2x}}-\frac{1}{4e^{2x}} + C, C \in \mathbb{R}\) Calculate the integral \( x^{2} \log x \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x^{2}\log_{10}{\left(x\right)} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=\log_{10}{\left(x\right)}\\&dv=x^{2}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=\frac{1}{\ln{\left(10\right)}\times x}dx\\&dv=x^{2}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=\frac{1}{\ln{\left(10\right)}\times x}dx\\&v=\frac{x^{3}}{3}\end{align}\) - step4: Substitute the values into formula: \(\log_{10}{\left(x\right)}\times \frac{x^{3}}{3}-\int \frac{1}{\ln{\left(10\right)}\times x}\times \frac{x^{3}}{3} dx\) - step5: Calculate: \(\frac{x^{3}\log_{10}{\left(x\right)}}{3}-\int \frac{x^{2}}{3\ln{\left(10\right)}} dx\) - step6: Evaluate the integral: \(\frac{x^{3}\log_{10}{\left(x\right)}}{3}-\frac{x^{3}}{9\ln{\left(10\right)}}\) - step7: Simplify the expression: \(\frac{1}{3}x^{3}\log_{10}{\left(x\right)}-\frac{x^{3}}{9\ln{\left(10\right)}}\) - step8: Simplify the expression: \(\frac{1}{3}x^{3}\log_{10}{\left(x\right)}-\frac{1}{9\ln{\left(10\right)}}\times x^{3}\) - step9: Add the constant of integral C: \(\frac{1}{3}x^{3}\log_{10}{\left(x\right)}-\frac{1}{9\ln{\left(10\right)}}\times x^{3} + C, C \in \mathbb{R}\) Calculate the integral \( x(3 x-5)^{4} \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int x\left(3x-5\right)^{4} dx\) - step1: Use the substitution \(dx=\frac{1}{3} dt\) to transform the integral\(:\) \(\int x\left(3x-5\right)^{4}\times \frac{1}{3} dt\) - step2: Simplify: \(\int \frac{\left(3x-5\right)^{4}}{3}x dt\) - step3: Use the substitution \(t=3x-5\) to transform the integral\(:\) \(\int \frac{t^{4}}{3}\left(\frac{1}{3}t+\frac{5}{3}\right) dt\) - step4: Simplify: \(\int \frac{\frac{1}{3}t^{5}+\frac{5}{3}t^{4}}{3} dt\) - step5: Rewrite the expression: \(\int \frac{1}{3}\left(\frac{1}{3}t^{5}+\frac{5}{3}t^{4}\right) dt\) - step6: Use properties of integrals: \(\frac{1}{3}\times \int \left(\frac{1}{3}t^{5}+\frac{5}{3}t^{4}\right) dt\) - step7: Use properties of integrals: \(\frac{1}{3}\left(\int \frac{1}{3}t^{5} dt+\int \frac{5}{3}t^{4} dt\right)\) - step8: Calculate: \(\frac{1}{3}\times \int \frac{1}{3}t^{5} dt+\frac{1}{3}\times \int \frac{5}{3}t^{4} dt\) - step9: Evaluate the integral: \(\frac{t^{6}}{54}+\frac{1}{3}\times \int \frac{5}{3}t^{4} dt\) - step10: Evaluate the integral: \(\frac{t^{6}}{54}+\frac{t^{5}}{9}\) - step11: Substitute back: \(\frac{\left(3x-5\right)^{6}}{54}+\frac{\left(3x-5\right)^{5}}{9}\) - step12: Evaluate the power: \(\frac{\left(3x-5\right)^{6}}{54}+\frac{243x^{5}-2025x^{4}+6750x^{3}-11250x^{2}+9375x-3125}{9}\) - step13: Add the constant of integral C: \(\frac{\left(3x-5\right)^{6}}{54}+\frac{243x^{5}-2025x^{4}+6750x^{3}-11250x^{2}+9375x-3125}{9} + C, C \in \mathbb{R}\) Here are the solutions for the integrals provided: 1. **Integral of \(x^{2}+3x-2\)**: \[ \int (x^{2}+3x-2) \, dx = \frac{x^{3}}{3} + \frac{3x^{2}}{2} - 2x + C, \quad C \in \mathbb{R} \] 2. **Integral of \(\frac{1}{(3x+5)^{4}}\)**: \[ \int \frac{1}{(3x+5)^{4}} \, dx = -\frac{1}{9(3x+5)^{3}} + C, \quad C \in \mathbb{R} \] 3. **Integral of \(\frac{1}{(x+1)^{3}}\)**: \[ \int \frac{1}{(x+1)^{3}} \, dx = -\frac{1}{2(x+1)^{2}} + C, \quad C \in \mathbb{R} \] 4. **Integral of \(xe^{-2x}\)** (using integration by parts): \[ \int x e^{-2x} \, dx = -\frac{x}{2e^{2x}} - \frac{1}{4e^{2x}} + C, \quad C \in \mathbb{R} \] 5. **Integral of \(x(3x-5)^{4}\)**: \[ \int x(3x-5)^{4} \, dx = \frac{(3x-5)^{6}}{54} + \frac{243x^{5}-2025x^{4}+6750x^{3}-11250x^{2}+9375x-3125}{9} + C, \quad C \in \mathbb{R} \] 6. **Integral of \(x^{2}e^{x}\)** (using integration by parts): \[ \int x^{2} e^{x} \, dx = x^{2}e^{x} - 2xe^{x} + 2e^{x} + C, \quad C \in \mathbb{R} \] 7. **Integral of \(x^{2} \log x\)** (using integration by parts): \[ \int x^{2} \log x \, dx = \frac{1}{3}x^{3}\log_{10}(x) - \frac{1}{9\ln(10)}x^{3} + C, \quad C \in \mathbb{R} \] 8. **Integral of \(4^{x} e^{2x}\)**: \[ \int 4^{x} e^{2x} \, dx = \frac{(4e^{2})^{x}}{2\ln(2e)} + C, \quad C \in \mathbb{R} \] These results provide the indefinite integrals for each of the expressions given.