Indefinite integral via u-substitution

Let $u = x^2+1$. Then $du = 2x\,dx$, which matches the factor $2x\,dx$ already present in the integral. Substituting gives $\int u^5\,du$.

Integrating, $\int u^5\,du = \frac{u^6}{6} + C$. Replacing $u$ with $x^2+1$ gives the final answer $\frac{(x^2+1)^6}{6} + C$.

The substitution was u equals x squared plus one. This was chosen because its derivative, two x, appeared exactly as a factor in the original integral. That made the integral simplify directly into a power of u, which is easy to integrate.