If ∫ x⁵ cos(x⁶) dx = k sin(x⁶) + C, find the value of k.

Put $x^6 = t$, so $6x^5\,dx = dt$ and $x^5\,dx = \frac{dt}{6}$. $\int x^5\cos(x^6)\,dx = \frac{1}{6}\int \cos t\,dt = \frac{1}{6}\sin t + C = \frac{1}{6}\sin(x^6) + C$ Comparing with $k\sin(x^6) + C$: $k = \frac{1}{6}$.