1. \( \int x \ln x d x \). 2. \( \int x e^{3 x} d x \). 3. \( \int x^{3} e^{x^{2}} d x \). 4. \( \int \cos \sqrt{x} d x \). 5. \( \int \frac{x e^{x}}{(x+1)^{2}} d x \).

Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int \frac{xe^{x}}{\left(x+1\right)^{2}} dx\) - step1: Evaluate the integral: \(\int \left(x+1\right)^{-2}xe^{x} dx\) - step2: Prepare for integration by parts: \(\begin{align}&u=xe^{x}\\&dv=\left(x+1\right)^{-2}dx\end{align}\) - step3: Calculate the derivative: \(\begin{align}&du=\left(e^{x}+xe^{x}\right)dx\\&dv=\left(x+1\right)^{-2}dx\end{align}\) - step4: Evaluate the integral: \(\begin{align}&du=\left(e^{x}+xe^{x}\right)dx\\&v=\frac{\left(x+1\right)^{-1}}{-1}\end{align}\) - step5: Substitute the values into formula: \(xe^{x}\times \frac{\left(x+1\right)^{-1}}{-1}-\int \left(e^{x}+xe^{x}\right)\times \frac{\left(x+1\right)^{-1}}{-1} dx\) - step6: Calculate: \(-xe^{x}\left(x+1\right)^{-1}-\int -e^{x} dx\) - step7: Evaluate the integral: \(-xe^{x}\left(x+1\right)^{-1}+e^{x}\) - step8: Multiply the terms: \(-\frac{xe^{x}}{x+1}+e^{x}\) - step9: Add the constant of integral C: \(-\frac{xe^{x}}{x+1}+e^{x} + C, C \in \mathbb{R}\) Calculate the integral \( \int x \ln x \, dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x\ln{\left(x\right)} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=\ln{\left(x\right)}\\&dv=xdx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=\frac{1}{x}dx\\&dv=xdx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=\frac{1}{x}dx\\&v=\frac{x^{2}}{2}\end{align}\) - step4: Substitute the values into formula: \(\ln{\left(x\right)}\times \frac{x^{2}}{2}-\int \frac{1}{x}\times \frac{x^{2}}{2} dx\) - step5: Calculate: \(\frac{x^{2}\ln{\left(x\right)}}{2}-\int \frac{x}{2} dx\) - step6: Evaluate the integral: \(\frac{x^{2}\ln{\left(x\right)}}{2}-\frac{x^{2}}{4}\) - step7: Simplify the expression: \(\frac{1}{2}x^{2}\ln{\left(x\right)}-\frac{x^{2}}{4}\) - step8: Add the constant of integral C: \(\frac{1}{2}x^{2}\ln{\left(x\right)}-\frac{x^{2}}{4} + C, C \in \mathbb{R}\) Calculate the integral \( \int x e^{3 x} \, dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int xe^{3x} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=e^{3x}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=dx\\&dv=e^{3x}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=dx\\&v=\frac{e^{3x}}{3}\end{align}\) - step4: Substitute the values into formula: \(x\times \frac{e^{3x}}{3}-\int 1\times \frac{e^{3x}}{3} dx\) - step5: Calculate: \(\frac{xe^{3x}}{3}-\int \frac{e^{3x}}{3} dx\) - step6: Evaluate the integral: \(\frac{xe^{3x}}{3}-\frac{e^{3x}}{9}\) - step7: Simplify the expression: \(\frac{e^{3x}x}{3}-\frac{e^{3x}}{9}\) - step8: Add the constant of integral C: \(\frac{e^{3x}x}{3}-\frac{e^{3x}}{9} + C, C \in \mathbb{R}\) Calculate the integral \( \int x^{3} e^{x^{2}} \, dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int x^{3}e^{\left(x^{2}\right)} dx\) - step1: Use the substitution \(dx=\frac{1}{2x} dt\) to transform the integral\(:\) \(\int x^{3}e^{\left(x^{2}\right)}\times \frac{1}{2x} dt\) - step2: Simplify: \(\int \frac{e^{\left(x^{2}\right)}x^{2}}{2} dt\) - step3: Use the substitution \(t=x^{2}\) to transform the integral\(:\) \(\int \frac{e^{t}t}{2} dt\) - step4: Rewrite the expression: \(\int \frac{1}{2}e^{t}t dt\) - step5: Use properties of integrals: \(\frac{1}{2}\times \int e^{t}t dt\) - step6: Prepare for integration by parts: \(\begin{align}&u=t\\&dv=e^{t}dt\end{align}\) - step7: Calculate the derivative: \(\begin{align}&du=dt\\&dv=e^{t}dt\end{align}\) - step8: Evaluate the integral: \(\begin{align}&du=dt\\&v=e^{t}\end{align}\) - step9: Substitute the values into formula: \(\frac{1}{2}\left(te^{t}-\int 1\times e^{t} dt\right)\) - step10: Calculate: \(\frac{1}{2}\left(te^{t}-\int e^{t} dt\right)\) - step11: Calculate: \(\frac{1}{2}te^{t}-\frac{1}{2}\times \int e^{t} dt\) - step12: Evaluate the integral: \(\frac{1}{2}te^{t}-\frac{e^{t}}{2}\) - step13: Substitute back: \(\frac{1}{2}x^{2}e^{\left(x^{2}\right)}-\frac{e^{\left(x^{2}\right)}}{2}\) - step14: Rewrite the expression: \(\frac{e^{\left(x^{2}\right)}x^{2}}{2}-\frac{e^{\left(x^{2}\right)}}{2}\) - step15: Add the constant of integral C: \(\frac{e^{\left(x^{2}\right)}x^{2}}{2}-\frac{e^{\left(x^{2}\right)}}{2} + C, C \in \mathbb{R}\) Aquí están las soluciones a las integrales solicitadas: 1. Para \( \int x \ln x \, dx \): \[ \frac{1}{2}x^{2}\ln{(x)} - \frac{x^{2}}{4} + C, \quad C \in \mathbb{R} \] 2. Para \( \int x e^{3 x} \, dx \): \[ \frac{e^{3x}x}{3} - \frac{e^{3x}}{9} + C, \quad C \in \mathbb{R} \] 3. Para \( \int x^{3} e^{x^{2}} \, dx \): \[ \frac{e^{(x^{2})}x^{2}}{2} - \frac{e^{(x^{2})}}{2} + C, \quad C \in \mathbb{R} \] 4. Para \( \int \cos \sqrt{x} \, dx \), no se pudo calcular debido a que el método de sustitución no es compatible con esta integral. 5. Para \( \int \frac{x e^{x}}{(x+1)^{2}} \, dx \): \[ -\frac{xe^{x}}{x+1} + e^{x} + C, \quad C \in \mathbb{R} \] Si necesitas más ayuda con la integral que no se pudo calcular o cualquier otra consulta, házmelo saber.