91Ó°ÊÓ

The substitution method is a powerful technique used to simplify complex integrals. When faced with an integral that seems difficult to solve directly, substitution can make it more manageable.

In this approach, we introduce a new variable, usually denoted as \( u \), to replace a portion of the integrand. This helps convert the integral into a simpler form.

For example, in the original problem \( u = a + bx \) was chosen. This is a strategic choice because it replaces a complex expression with a simpler variable \( u \). The main task is to transform \( dx \) into terms of \( du \), achieved by finding \( du \). Here:

• We differentiate \( u = a + bx \), resulting in \( du = b \, dx \).
• Then solve for \( dx \), so \( dx = \frac{du}{b} \).

This substitution changes the original integral in terms of \( x \) into an integral in terms of \( u \), simplifying the problem significantly.

The power integration rule is a fundamental tool in calculus, making the integration of polynomials straightforward. This rule deals with functions of the form \( x^m \).

The general formula is: \[ \int x^m \, dx = \frac{x^{m+1}}{m+1} + C, \] where \( m eq -1 \). This formula originates from reversing the process of differentiation for polynomials.

In the solved exercise, after substituting \( u \) for \( a + bx \), the integral was:

• \( \,\int u^{\frac{1}{n}} \, du \), which mirrors \( x^m \) with \( m = \frac{1}{n} \).
• Plug \( m = \frac{1}{n} \) into the power integration rule:
  
  \[ \int u^{\frac{1}{n}} \, du = \frac{u^{\frac{1+n}{n}}}{\frac{1+n}{n}} + C. \]

This step transforms the integral into a form that can easily be evaluated.

Indefinite integrals represent the family of all antiderivatives of a function. Unlike definite integrals, they do not have upper and lower limits.

The result of an indefinite integral is a general expression plus a constant \( C \). This constant ensures all possible antiderivatives are covered because differentiation introduces potential constants.

In the example problem, after integration, we have:

• \( \frac{n}{b(n+1)} (a+bx)^{\frac{n+1}{n}} + C \)

Key parts of this expression:

• The term \( (a+bx)^{\frac{n+1}{n}} \) shows the integration result of \( u \) back-substituted to \( x \).
• \( \frac{n}{b(n+1)} \) is the factor from the substitution and power rule application.

The presence of \( C \) indicates all possible antiderivatives are accounted for, demonstrating the essence of indefinite integrals.