Solution: Let $ \sqrt{u} = x $, so $ u = x^2 $. Substituting into the equation gives:

["Solution: Let ( \sqrt{u} = x ), so ( u = x^2 ). Substituting into the equation gives a powerful method for simplifying square root expressions.", "When tackling radical equations or complex integrals involving square roots, substitution is one of the most effective strategies. A clean and elegant approach is letting ( \sqrt{u} = x ), which instantly transforms the radical expression into a polynomial form—making it much easier to manipulate, solve, or differentiate.", "Why Substitution Works\nStarting with ( \sqrt{u} = x ), we immediately have:\n[\nu = x^2\n]\nThis substitution eliminates the square root, replacing it with a quadratic term. The replacement is especially valuable when working with integrals, derivatives, or algebraic expressions containing ( \sqrt{u} ). Instead of dealing with irrational numbers or nested radicals, you work with a simpler ( x )-based form.", "Applying the Substitution\nSuppose the original equation involves ( \sqrt{u} ), like:\n[\n\int \sqrt{u} \cdot (2u + 3) , du\n]\nBy substituting ( u = x^2 ) and ( \sqrt{u} = x ), we also need ( du ):\n[\ndu = 2x , dx\n]\nNow the integral becomes:\n[\n\int x \cdot (2x^2 + 3) \cdot 2x , dx = \int x(2x^2 + 3) \cdot 2x , dx = \int 2x^2(2x^2 + 3) , dx\n]\n[\n= \int (4x^4 + 6x^2) , dx\n]\nThis polynomial expression is straightforward to integrate using basic power rule techniques.", "Advantages of This Technique\n- Removes radical complexity by conversion to polynomials.\n- Simplifies both integration and differentiation of radical-based functions.\n- Makes algebraic manipulation more transparent, especially in calculus and equation solving.\n- Useful in physics, engineering, and applied mathematics for modeling systems with square root dependencies.", "Conclusion\nLet ( \sqrt{u} = x \Rightarrow u = x^2 ) is more than a substitution—it’s a powerful algebraic transformation that unlocks easier computation and insight. Whether solving integrals, solving equations, or calculus problems involving radicals, this method streamlines the process with minimal effort and maximum clarity.", "Keywords:\nsolve ( \sqrt{u} ) substitution, ( u = x^2 \ method, radical equation solution, calculus substitution, integral with square roots, ( \sqrt{u} \ to x, algebraic simplification, substitution in calculus.\nMeta Description:\nDiscover how letting ( \sqrt{u} = x ) and using ( u = x^2 ) simplifies solving radical equations and integrals. Learn the step-by-step substitution technique for easier calculus and algebra."]