Integrate the following functions using the appropriate rules of integration. a. \( \int_{1}^{2} x \ln (x) d x \) (5 Marks) b. \( \int_{0}^{2} x\left(x^{2}-1\right)^{7} d x \) (7 Marks)

Calculate the integral \( \int_{1}^{2} x \ln (x) d x \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int_{1}^{2} x\ln{\left(x\right)} dx\) - step1: Evaluate the integral: \(\int x\ln{\left(x\right)} dx\) - step2: Prepare for integration by parts: \(\begin{align}&u=\ln{\left(x\right)}\\&dv=xdx\end{align}\) - step3: Calculate the derivative: \(\begin{align}&du=\frac{1}{x}dx\\&dv=xdx\end{align}\) - step4: Evaluate the integral: \(\begin{align}&du=\frac{1}{x}dx\\&v=\frac{x^{2}}{2}\end{align}\) - step5: Substitute the values into formula: \(\ln{\left(x\right)}\times \frac{x^{2}}{2}-\int \frac{1}{x}\times \frac{x^{2}}{2} dx\) - step6: Calculate: \(\frac{x^{2}\ln{\left(x\right)}}{2}-\int \frac{x}{2} dx\) - step7: Evaluate the integral: \(\frac{x^{2}\ln{\left(x\right)}}{2}-\frac{x^{2}}{4}\) - step8: Simplify the expression: \(\frac{1}{2}x^{2}\ln{\left(x\right)}-\frac{x^{2}}{4}\) - step9: Return the limits: \(\left(\frac{1}{2}x^{2}\ln{\left(x\right)}-\frac{x^{2}}{4}\right)\bigg |_{1}^{2}\) - step10: Calculate the value: \(2\ln{\left(2\right)}-\frac{3}{4}\) - step11: Calculate: \(\frac{8\ln{\left(2\right)}-3}{4}\) Calculate the integral \( \int_{0}^{2} x\left(x^{2}-1\right)^{7} d x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int_{0}^{2} x\left(x^{2}-1\right)^{7} dx\) - step1: Evaluate the integral: \(\int x\left(x^{2}-1\right)^{7} dx\) - step2: Use the substitution \(dx=\frac{1}{2x} dt\) to transform the integral\(:\) \(\int x\left(x^{2}-1\right)^{7}\times \frac{1}{2x} dt\) - step3: Simplify: \(\int \frac{\left(x^{2}-1\right)^{7}}{2} dt\) - step4: Use the substitution \(t=x^{2}\) to transform the integral\(:\) \(\int \frac{\left(t-1\right)^{7}}{2} dt\) - step5: Rewrite the expression: \(\int \frac{1}{2}\left(t-1\right)^{7} dt\) - step6: Use properties of integrals: \(\frac{1}{2}\times \int \left(t-1\right)^{7} dt\) - step7: Use the substitution \(dt=1 dv\) to transform the integral\(:\) \(\frac{1}{2}\times \int \left(t-1\right)^{7}\times 1 dv\) - step8: Simplify: \(\frac{1}{2}\times \int \left(t-1\right)^{7} dv\) - step9: Use the substitution \(v=t-1\) to transform the integral\(:\) \(\frac{1}{2}\times \int v^{7} dv\) - step10: Evaluate the integral: \(\frac{1}{2}\times \frac{v^{7+1}}{7+1}\) - step11: Simplify: \(\frac{1}{2}\times \frac{v^{8}}{8}\) - step12: Multiply the terms: \(\frac{v^{8}}{2\times 8}\) - step13: Multiply the terms: \(\frac{v^{8}}{16}\) - step14: Substitute back: \(\frac{\left(t-1\right)^{8}}{16}\) - step15: Substitute back: \(\frac{\left(x^{2}-1\right)^{8}}{16}\) - step16: Return the limits: \(\left(\frac{\left(x^{2}-1\right)^{8}}{16}\right)\bigg |_{0}^{2}\) - step17: Calculate the value: \(410\) The integral of \( \int_{1}^{2} x \ln (x) d x \) is \( \frac{8\ln{(2)}-3}{4} \). The integral of \( \int_{0}^{2} x\left(x^{2}-1\right)^{7} d x \) is 410.