Evaluate \( \int\left(2 x^{3}+1\right)^{7} x^{2} d x \)

Calculate the integral \( \int\left(2 x^{3}+1\right)^{7} x^{2} dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \left(2x^{3}+1\right)^{7}x^{2} dx\) - step1: Use the substitution \(dx=\frac{1}{6x^{2}} dt\) to transform the integral\(:\) \(\int \left(2x^{3}+1\right)^{7}x^{2}\times \frac{1}{6x^{2}} dt\) - step2: Simplify: \(\int \frac{\left(2x^{3}+1\right)^{7}}{6} dt\) - step3: Use the substitution \(t=2x^{3}\) to transform the integral\(:\) \(\int \frac{\left(t+1\right)^{7}}{6} dt\) - step4: Use properties of integrals: \(\frac{1}{6}\times \int \left(t+1\right)^{7} dt\) - step5: Use the substitution \(dt=1 dv\) to transform the integral\(:\) \(\frac{1}{6}\times \int \left(t+1\right)^{7}\times 1 dv\) - step6: Simplify: \(\frac{1}{6}\times \int \left(t+1\right)^{7} dv\) - step7: Use the substitution \(v=t+1\) to transform the integral\(:\) \(\frac{1}{6}\times \int v^{7} dv\) - step8: Evaluate the integral: \(\frac{1}{6}\times \frac{v^{7+1}}{7+1}\) - step9: Simplify: \(\frac{1}{6}\times \frac{v^{8}}{8}\) - step10: Multiply the terms: \(\frac{v^{8}}{6\times 8}\) - step11: Multiply the terms: \(\frac{v^{8}}{48}\) - step12: Substitute back: \(\frac{\left(t+1\right)^{8}}{48}\) - step13: Substitute back: \(\frac{\left(2x^{3}+1\right)^{8}}{48}\) - step14: Add the constant of integral C: \(\frac{\left(2x^{3}+1\right)^{8}}{48} + C, C \in \mathbb{R}\) The result of the integral \( \int\left(2 x^{3}+1\right)^{7} x^{2} dx \) is \( \frac{(2x^{3}+1)^{8}}{48} + C \), where \( C \) is an arbitrary constant.