Let’s analyze and solve the given integral step by step.

The problem is to evaluate the definite integral:

$\int_0^1 \frac{(\sqrt{x} + 1)(x^2 + \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}} \, dx$

Step 1: Simplify the denominator

The denominator is: $x \sqrt{x} + x + \sqrt{x}$ Factoring $\sqrt{x}$ out: $x \sqrt{x} + x + \sqrt{x} = \sqrt{x} (x + 1) + \sqrt{x} = \sqrt{x} (x + 1) + 1$ This simplifies the expression.

Step 2: Substitution and simplification

Let me perform calculations systematically. sympy import symbols, sqrt, integrate

Define the variable

x = symbols('x')

Define the integral's numerator and denominator

numerator = (sqrt(x) + 1) * (x**2 + sqrt(x)) denominator = x * sqrt(x) + x + sqrt(x)

Simplify the denominator

simplified_denominator = denominator.simplify()

Define the full integrand

integrand = numerator / simplified_denominator

Compute the definite integral

result = integrate(integrand, (x, 0, 1)) resultThe value of the definite integral is:

$\boxed{\frac{7}{2} - \frac{4\sqrt{3}\pi}{9}}$

Let me know if you'd like me to break this down further or provide more explanation!

Would you like to:

1. Understand the steps in detail, including the algebraic simplifications?
2. See a graphical representation of the function and its integral?
3. Explore substitutions used for similar integrals?
4. Derive the result numerically to confirm the answer?
5. Learn techniques for tackling integrals with square roots?

Tip: Simplify expressions as much as possible before integrating—this can save a lot of computation time!