(c) Evaluate the following definite integrals: (i) \( \int_{4}^{5}\left(\frac{2}{\sqrt{x}}-x\right) d x \) (ii) \( \int_{0}^{1} \frac{x}{\left(2 x^{2}+1\right)^{3}} d x \)

Calculate the integral \( \int_{4}^{5}\left(\frac{2}{\sqrt{x}}-x\right) dx \). Evaluate the integral by following steps: - step0: Evaluate: \(\int_{4}^{5} \frac{2}{\sqrt{x}}-x dx\) - step1: Evaluate the power: \(\int_{4}^{5} \frac{2}{x^{\frac{1}{2}}}-x dx\) - step2: Evaluate the integral: \(\int \frac{2}{x^{\frac{1}{2}}}-x dx\) - step3: Use properties of integrals: \(\int \frac{2}{x^{\frac{1}{2}}} dx-\int x dx\) - step4: Evaluate the integral: \(4x^{\frac{1}{2}}-\frac{1}{2}x^{2}\) - step5: Return the limits: \(\left(4x^{\frac{1}{2}}-\frac{1}{2}x^{2}\right)\bigg |_{4}^{5}\) - step6: Calculate the value: \(4\times 5^{\frac{1}{2}}-\frac{25}{2}\) - step7: Transform the expression: \(4\sqrt{5}-\frac{25}{2}\) - step8: Calculate: \(\frac{8\sqrt{5}-25}{2}\) Calculate the integral \( \int_{0}^{1} \frac{x}{(2x^{2}+1)^{3}} dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int_{0}^{1} \frac{x}{\left(2x^{2}+1\right)^{3}} dx\) - step1: Evaluate the integral: \(\int \frac{x}{\left(2x^{2}+1\right)^{3}} dx\) - step2: Use the substitution \(dx=\frac{1}{2x} dt\) to transform the integral\(:\) \(\int \frac{x}{\left(2x^{2}+1\right)^{3}}\times \frac{1}{2x} dt\) - step3: Simplify: \(\int \frac{1}{2\left(2x^{2}+1\right)^{3}} dt\) - step4: Use the substitution \(t=x^{2}\) to transform the integral\(:\) \(\int \frac{1}{2\left(2t+1\right)^{3}} dt\) - step5: Use properties of integrals: \(\frac{1}{2}\times \int \frac{1}{\left(2t+1\right)^{3}} dt\) - step6: Use the substitution \(v=2t+1\) to transform the integral\(:\) \(\frac{1}{2}\times \int \frac{1}{2v^{3}} dv\) - step7: Use properties of integrals: \(\frac{1}{4}\times \int \frac{1}{v^{3}} dv\) - step8: Evaluate the integral: \(\frac{1}{4}\left(-\frac{1}{2v^{2}}\right)\) - step9: Calculate: \(-\frac{1}{8v^{2}}\) - step10: Substitute back: \(-\frac{1}{8\left(2t+1\right)^{2}}\) - step11: Substitute back: \(-\frac{1}{8\left(2x^{2}+1\right)^{2}}\) - step12: Return the limits: \(\left(-\frac{1}{8\left(2x^{2}+1\right)^{2}}\right)\bigg |_{0}^{1}\) - step13: Calculate the value: \(\frac{1}{9}\) The results of the definite integrals are as follows: (i) \( \int_{4}^{5}\left(\frac{2}{\sqrt{x}}-x\right) dx = \frac{8\sqrt{5}-25}{2} \) (ii) \( \int_{0}^{1} \frac{x}{(2x^{2}+1)^{3}} dx = \frac{1}{9} \) or \( 0.\dot{1} \)