Find the indefinite integral. \( \int \frac{x}{\sqrt[3]{x+10}} d x \) \( \int \frac{x}{\sqrt[3]{x+10}} d x=\square \)

Calculate the integral \( \int \frac{x}{\sqrt[3]{x+10}} dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int \frac{x}{\sqrt[3]{x+10}} dx\) - step1: Evaluate the power: \(\int \frac{x}{\left(x+10\right)^{\frac{1}{3}}} dx\) - step2: Evaluate the integral: \(\int x\left(x+10\right)^{-\frac{1}{3}} dx\) - step3: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=\left(x+10\right)^{-\frac{1}{3}}dx\end{align}\) - step4: Calculate the derivative: \(\begin{align}&du=dx\\&dv=\left(x+10\right)^{-\frac{1}{3}}dx\end{align}\) - step5: Evaluate the integral: \(\begin{align}&du=dx\\&v=\frac{3}{2}\left(x+10\right)^{\frac{2}{3}}\end{align}\) - step6: Substitute the values into formula: \(x\times \frac{3}{2}\left(x+10\right)^{\frac{2}{3}}-\int 1\times \frac{3}{2}\left(x+10\right)^{\frac{2}{3}} dx\) - step7: Calculate: \(\frac{3}{2}x\left(x+10\right)^{\frac{2}{3}}-\int \frac{3}{2}\left(x+10\right)^{\frac{2}{3}} dx\) - step8: Evaluate the integral: \(\frac{3}{2}x\left(x+10\right)^{\frac{2}{3}}-\frac{9\left(x+10\right)^{\frac{5}{3}}}{10}\) - step9: Simplify: \(\frac{3}{2}x\sqrt[3]{\left(x+10\right)^{2}}-\frac{9\left(x+10\right)^{\frac{5}{3}}}{10}\) - step10: Simplify: \(\frac{3}{2}x\sqrt[3]{\left(x+10\right)^{2}}-\frac{9\sqrt[3]{\left(x+10\right)^{5}}}{10}\) - step11: Expand the expression: \(\frac{3}{2}x\sqrt[3]{\left(x+10\right)^{2}}-\frac{9x\sqrt[3]{\left(x+10\right)^{2}}+90\sqrt[3]{\left(x+10\right)^{2}}}{10}\) - step12: Add the constant of integral C: \(\frac{3}{2}x\sqrt[3]{\left(x+10\right)^{2}}-\frac{9x\sqrt[3]{\left(x+10\right)^{2}}+90\sqrt[3]{\left(x+10\right)^{2}}}{10} + C, C \in \mathbb{R}\) The indefinite integral of \( \int \frac{x}{\sqrt[3]{x+10}} dx \) is \( \frac{3}{2}x\sqrt[3]{(x+10)^{2}}-\frac{9x\sqrt[3]{(x+10)^{2}}+90\sqrt[3]{(x+10)^{2}}}{10} + C \), where \( C \) is an arbitrary constant.