\( \int \sqrt[ 3 ] { x - 3 } d x = \ldots \)

Calculate the integral \( \int \sqrt[3]{x-3}dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \sqrt[3]{x-3} dx\) - step1: Evaluate the power: \(\int \left(x-3\right)^{\frac{1}{3}} dx\) - step2: Use the substitution \(dx=1 dt\) to transform the integral\(:\) \(\int \left(x-3\right)^{\frac{1}{3}}\times 1 dt\) - step3: Simplify: \(\int \left(x-3\right)^{\frac{1}{3}} dt\) - step4: Use the substitution \(t=x-3\) to transform the integral\(:\) \(\int t^{\frac{1}{3}} dt\) - step5: Evaluate the integral: \(\frac{t^{\frac{1}{3}+1}}{\frac{1}{3}+1}\) - step6: Add the numbers: \(\frac{t^{\frac{4}{3}}}{\frac{1}{3}+1}\) - step7: Add the numbers: \(\frac{t^{\frac{4}{3}}}{\frac{4}{3}}\) - step8: Multiply by the reciprocal: \(t^{\frac{4}{3}}\times \frac{3}{4}\) - step9: Reorder the terms: \(\frac{3}{4}t^{\frac{4}{3}}\) - step10: Substitute back: \(\frac{3}{4}\left(x-3\right)^{\frac{4}{3}}\) - step11: Simplify: \(\frac{3}{4}\sqrt[3]{\left(x-3\right)^{4}}\) - step12: Add the constant of integral C: \(\frac{3}{4}\sqrt[3]{\left(x-3\right)^{4}} + C, C \in \mathbb{R}\) The integral of \( \sqrt[3]{x-3} \) with respect to \( x \) is \( \frac{3}{4}\sqrt[3]{(x-3)^4} + C \), where \( C \) is an arbitrary constant.