91Ó°ÊÓ

The substitution method is a fundamental technique when dealing with integrals, especially improper ones. This method transforms a complex integral into a simpler one by changing variables, making it easier to solve. It's similar to the chain rule in differentiation. By setting a part of the integrand to a new variable, we can simplify calculations.
In the given exercise, we used the substitution \( u = p^2 + 1 \), which resulted in \( du = 2p \, dp \). This substitution helped change the original integral into a more manageable form: \( \frac{1}{2} \int \frac{1}{\sqrt[5]{u}} \, du \).
Keep in mind that with substitution, you also need to adjust the limits of integration. Initially, the limits were from \( p = 0 \) to \( p = \infty \). After substitution, the limits became \( u = 1 \) to \( u = \infty \), reflecting the change in variables. This adjustment ensures that our evaluation aligns with the transformed integral.

Integral convergence is a crucial concept to determine whether an integral has a finite value or not. An improper integral, like the exercise one with infinite limits, tests convergence.
When evaluating an integral like \( \int_{0}^{\infty} \), we need to check whether it converges and has a finite limit, or diverges, meaning it grows without bound.

• To determine convergence, examine the behavior of the function as it approaches the limits of integration.
• The criteria used involve the exponent \( r \) in the expression \( \int_{a}^{\infty} x^r \, dx \). If \( -1 < r < 0 \), the integral converges.

In the given exercise, our transformed integral \( \frac{1}{2} \int_{1}^{\infty} u^{-\frac{1}{5}} \, du \) has \( r = -\frac{1}{5} \), which ensures convergence as it falls within the specified range. Thus, the integral converges but does not equal a finite number, implying divergence in practical sense.

The power rule for integration allows us to easily integrate functions of the form \( x^n \). This rule simplifies finding antiderivatives, crucial for evaluating integrals after substitution.
The formula used is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \), where \( C \) is the constant of integration.

• It's used to evaluate parts of an integrand that follow this power pattern.
• The power rule is applied after using substitution to simplify the expression.

For the exercise, after substituting, we applied the power rule: \( \frac{1}{2}\left[ \frac{5}{4} u^{\frac{4}{5}} \right]_{1}^{\infty} \).
Evaluating the antiderivative involved using the formula with the limits substituted for \( u \). The rule immensely simplifies the integral solving process, ensuring our results are exact.