Find the integral \( \int x^{5} \cos \left(x^{3}\right) d x= \)

Calculate the integral \( \int x^{5} \cos \left(x^{3}\right) d x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int x^{5}\cos\left(x^{3}\right) dx\) - step1: Use the substitution \(dx=\frac{1}{3x^{2}} dt\) to transform the integral\(:\) \(\int x^{5}\cos\left(x^{3}\right)\times \frac{1}{3x^{2}} dt\) - step2: Simplify: \(\int \frac{x^{3}\cos\left(x^{3}\right)}{3} dt\) - step3: Use the substitution \(t=x^{3}\) to transform the integral\(:\) \(\int \frac{t\cos\left(t\right)}{3} dt\) - step4: Simplify the expression: \(\int \frac{1}{3}t\cos\left(t\right) dt\) - step5: Use properties of integrals: \(\frac{1}{3}\times \int t\cos\left(t\right) dt\) - step6: Prepare for integration by parts: \(\begin{align}&u=t\\&dv=\cos\left(t\right)dt\end{align}\) - step7: Calculate the derivative: \(\begin{align}&du=dt\\&dv=\cos\left(t\right)dt\end{align}\) - step8: Evaluate the integral: \(\begin{align}&du=dt\\&v=\sin\left(t\right)\end{align}\) - step9: Substitute the values into formula: \(\frac{1}{3}\left(t\sin\left(t\right)-\int 1\times \sin\left(t\right) dt\right)\) - step10: Calculate: \(\frac{1}{3}\left(t\sin\left(t\right)-\int \sin\left(t\right) dt\right)\) - step11: Calculate: \(\frac{1}{3}t\sin\left(t\right)-\frac{1}{3}\times \int \sin\left(t\right) dt\) - step12: Evaluate the integral: \(\frac{1}{3}t\sin\left(t\right)+\frac{1}{3}\cos\left(t\right)\) - step13: Substitute back: \(\frac{1}{3}x^{3}\sin\left(x^{3}\right)+\frac{1}{3}\cos\left(x^{3}\right)\) - step14: Add the constant of integral C: \(\frac{1}{3}x^{3}\sin\left(x^{3}\right)+\frac{1}{3}\cos\left(x^{3}\right) + C, C \in \mathbb{R}\) The integral of \(x^{5} \cos \left(x^{3}\right)\) with respect to \(x\) is \(\frac{1}{3}x^{3}\sin(x^{3}) + \frac{1}{3}\cos(x^{3}) + C\), where \(C\) is an arbitrary constant.