\( \int _{}^{}x^{4}\sqrt {x^{2}+1} d x \)

To solve \( \int x^{4}\sqrt{x^{2}+1} \, dx \), you can use integration by parts or substitution. One effective method is to use the substitution \( u = x^{2} + 1 \), which gives \( du = 2x \, dx \) or \( dx = \frac{du}{2\sqrt{u-1}} \). This reciprocal manipulation can still be tricky, but the expression can also lead you into standard forms. Furthermore, this integral often appears in physics or engineering contexts, especially when dealing with problems involving work against variable forces or calculating volumes of revolution. Understanding these applications can help relate abstract calculus concepts to tangible real-world scenarios!