d) \( \int \frac{\sqrt[5]{1+\sqrt{x}}}{\sqrt{x}} d x \)

Calculate or simplify the expression \( \int \frac{\sqrt[5]{1+\sqrt{x}}}{\sqrt{x}} dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{\sqrt[5]{1+\sqrt{x}}}{\sqrt{x}} dx\) - step1: Evaluate the power: \(\int \frac{\left(1+\sqrt{x}\right)^{\frac{1}{5}}}{\sqrt{x}} dx\) - step2: Rewrite the expression: \(\int \frac{\left(1+x^{\frac{1}{2}}\right)^{\frac{1}{5}}}{x^{\frac{1}{2}}} dx\) - step3: Use the substitution \(dx=2x^{\frac{1}{2}} dt\) to transform the integral\(:\) \(\int \frac{\left(1+x^{\frac{1}{2}}\right)^{\frac{1}{5}}}{x^{\frac{1}{2}}}\times 2x^{\frac{1}{2}} dt\) - step4: Simplify: \(\int 2\left(1+x^{\frac{1}{2}}\right)^{\frac{1}{5}} dt\) - step5: Use the substitution \(t=1+x^{\frac{1}{2}}\) to transform the integral\(:\) \(\int 2t^{\frac{1}{5}} dt\) - step6: Use properties of integrals: \(2\times \int t^{\frac{1}{5}} dt\) - step7: Evaluate the integral: \(2\times \frac{t^{\frac{1}{5}+1}}{\frac{1}{5}+1}\) - step8: Simplify: \(2\times \frac{5}{6}t^{\frac{6}{5}}\) - step9: Multiply the numbers: \(\frac{5}{3}t^{\frac{6}{5}}\) - step10: Substitute back: \(\frac{5}{3}\left(1+x^{\frac{1}{2}}\right)^{\frac{6}{5}}\) - step11: Simplify: \(\frac{5}{3}\sqrt[5]{\left(1+\sqrt{x}\right)^{6}}\) - step12: Add the constant of integral C: \(\frac{5}{3}\sqrt[5]{\left(1+\sqrt{x}\right)^{6}} + C, C \in \mathbb{R}\) La integral de \( \frac{\sqrt[5]{1+\sqrt{x}}}{\sqrt{x}} \) con respecto a \( x \) es \( \frac{5}{3}\sqrt[5]{(1+\sqrt{x})^{6}} + C \), donde \( C \) es una constante real.