91Ó°ÊÓ

The substitution method is a handy technique in calculus for simplifying integrals. It is particularly useful when dealing with complex expressions involving square roots, exponents, or other non-polynomial functions.

To use the substitution method, you choose a new variable, say \( u \), to replace a more complicated section of the integral. In the example provided, the expression \( 3t + 4 \) within the square root is chosen as \( u \). This makes the integration significantly easier.

Here’s a quick outline of the process:

• Identify a part of the integral to substitute, often simplified expressions inside other functions.
• Express the chosen function in terms of a new variable \( u \).
• Find the differential \( du \) related to the original variable \( dt \). In our example, \( du = 3 \, dt \) which implies \( dt = \frac{du}{3} \).
• Replace all instances of the original variable in the integral with expressions involving \( u \).

After substitution, you're left with a simpler integral to solve. When finished, substitute back the original variable to obtain the final solution.

Integrals in calculus can be definite or indefinite. Both play crucial roles in analyzing functions and areas under curves.

**Indefinite Integrals** reflect the inverse process of differentiation. An indefinite integral returns a family of functions, represented as the integral function plus an arbitrary constant \( C \). This constant arises since derivatives of functions differing only by a constant are identical. In our case, the final result adds \( C \) to account for this.

• Indefinite integrals are represented by the symbol \( \int ... \, dt \), without bounds.
• The goal is to find the antiderivative of the function.

**Definite Integrals**, on the other hand, are used to evaluate the area under a curve within specified limits. This involves finding the difference between the antiderivative at the upper and lower limits. In our exercise, we dealt only with an indefinite integral, focusing primarily on simplification and integration.

The power rule for integration is a fundamental tool in solving integrals involving polynomials. It provides a straight-forward way to integrate functions of the form \( x^n \). The rule states: if \( n eq -1 \), then the integral of \( x^n \) is given by \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]This rule, when applied correctly, allows for efficient computation of integrals, particularly when combined with techniques like substitution.

In the step-by-step solution shared, the power rule is applied after making the substitution \( u = 3t + 4 \). It addresses each term separately after the simplification:\( \sqrt{u} \) and \( \frac{4}{\sqrt{u}} \).

• \( \int u^{1/2} du \) gives \( \frac{2}{3} u^{3/2} \)
• \( \int u^{-1/2} du \) results in \( 2u^{1/2} \)

By adding \( C \), the constant of integration, the solution becomes complete, accounting for any constants lost in the process of differentiation.