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

To solve the integral \( \int x^{5} \cos \left(x^{3}\right) d x \), we can use the method of integration by parts or substitution. Let's use the substitution method. Let \( u = x^{3} \). Then, we have: \[ du = 3x^{2} dx \quad \Rightarrow \quad dx = \frac{du}{3x^{2}}. \] In terms of \( u \), we can express \( x^5 \) as: \[ x^5 = x^{3} \cdot x^{2} = u \cdot x^{2}. \] Now we need to express \( x^{2} \) in terms of \( u \). From our substitution: \[ x^{2} = \left(u^{1/3}\right)^{2} = u^{2/3}. \] Now, putting it all together, we have: \[ \int x^{5} \cos(x^{3}) dx = \int u \cdot u^{2/3} \cdot \frac{du}{3x^{2}} = \int \frac{u^{5/3}}{3 (\frac{u^{2/3}}{u^{1/3}})} du = \frac{1}{3} \int u^{5/3} \cos(u) du. \] Next, we can integrate \( \int u^{5/3} \cos(u) du \) using integration by parts or recognize it leads to a solution involving special functions (like the Fresnel integrals) or further parts applied iteratively. However, for simple understanding, you can write: \[ \int x^{5} \cos(x^{3}) dx = \text{special function representation} + C. \] Thus, the final answer is expressed in terms of functions that might not have a clean elementary format.