5. \( \int \frac{(x+1)(x-2)}{\sqrt{x}} d x \) 6. \( \int\left(3 x-x^{2}\right)^{2} x^{3} d x \) 7. \( \int\left(\frac{3}{2 x^{2}}-\frac{5}{x^{3}}+\sqrt[3]{ } x^{2}\right) d x \) 8. \( \int\left(\frac{1}{2 x}-2\right) d x \) 9. \( \int\left(\frac{2}{3 x}+3\right) d x \) 10. \( \int\left(\frac{2}{\sqrt[4]{x}}-3\right) d x \)

Calculating integrals can often feel like solving a mystery—every step reveals more about the function's secrets! For instance, in the integral \( \int \frac{(x+1)(x-2)}{\sqrt{x}} d x \), breaking it down, you can simplify the numerator and exploit properties of exponents when working with \( \sqrt{x} \). This approach not only makes the calculation more manageable but also grounds your results in a more intuitive understanding of how the function behaves. When tackling polynomials like in \( \int (3x - x^{2})^{2} x^{3} d x \), consider expanding the squared term first. It often makes everything much clearer! Many people mistakenly try to integrate before simplifying the expression, which can lead to cumbersome calculations or confusing steps. Remember, as in life, taking things one step at a time often reveals the best path forward!