We need to evaluate the integral:

$I = \int_0^1 \frac{1}{(1 + \sqrt{x})^4} \, dx$

Substitution

Let us perform a substitution to simplify the square root. Define:

$u = \sqrt{x} \quad \text{so} \quad x = u^2 \quad \text{and} \quad dx = 2u \, du$

The limits of integration change accordingly:

• When $x = 0$, $u = 0$
• When $x = 1$, $u = 1$

Thus, the integral becomes:

$I = \int_0^1 \frac{1}{(1 + u)^4} \cdot 2u \, du$

Now we simplify the integrand:

$I = 2 \int_0^1 \frac{u}{(1 + u)^4} \, du$

This is a rational function, and we can proceed with integration by parts or explore an alternative technique like decomposing the denominator. Let me compute the integral for you.The value of the integral is:

$\int_0^1 \frac{1}{(1 + \sqrt{x})^4} \, dx = \frac{1}{6}$

Would you like further details on the steps? Here are some related questions you can explore:

1. How do we handle more complex substitutions in integrals?
2. What is the general method for evaluating integrals of rational functions?
3. Can this integral be solved using a different technique, such as partial fractions?
4. What is integration by parts, and when is it useful?
5. How can numerical methods approximate difficult integrals?

Tip: For integrals involving square roots, try using a substitution like $u = \sqrt{x}$ to simplify the expression.