\( \int \frac { \sqrt { x } } { 7 + \sqrt[ 4 ] { x } } d x \)

To tackle the integral \( \int \frac { \sqrt { x } } { 7 + \sqrt[ 4 ] { x } } d x \), we can use a substitution that simplifies the integrand. Let \( u = \sqrt[4]{x} \), which means that \( x = u^4 \) and \( dx = 4u^3 \, du \). This transforms the integral into: \[ \int \frac{u^2}{7 + u} \cdot 4u^3 \, du = 4 \int \frac{u^5}{7 + u} \, du \] This integral can now be tackled using polynomial long division or further substitution methods. Through careful algebraic manipulation and possibly breaking it into simpler parts, you can ultimately solve for the integral. Happy integrating! For practical purposes, you could also check the result using numerical integration methods or software when the algebra starts to feel overwhelming – it’s always good to confirm your work!