91Ó°ÊÓ

Integration by substitution is a powerful technique for evaluating integrals, especially when the integral is not straightforward to solve as it stands. The essence of substitution is to turn a complex expression into a simpler one by replacing variables. This is done by setting a part of the integral expression equal to a new variable, usually denoted as \( u \).

For example, in the exercise, we set \( u = 3t + 2 \). This substitution simplifies the integration process by transforming a complex polynomial function of \( t \) into a simpler form of \( u \).

• This involves changing the differential as well. When \( u = 3t + 2 \), the derivative \( \frac{du}{dt} = 3 \). Rewriting this in terms of \( dt \), we have \( dt = \frac{du}{3} \).
• Replacing \( t \) and \( dt \) with \( u \) and \( du \) transforms the integral into an easier form: \( \int u^{2.4} \frac{du}{3} \).

Polynomial integration is a straightforward technique for integrating polynomials of the form \( u^n \). The rule for integrating a function \( u^n \) is to increase the power by one and then divide by this new power.

For an expression like \( u^{2.4} \), the integral is found by applying this basic power rule:

• First, raise the power of \( u \) from 2.4 to 3.4.
• Next, divide by the new exponent, 3.4.

Therefore, the integral becomes \( \frac{u^{3.4}}{3.4} \). This step simplifies the original problem significantly, allowing us to solve the integral without directly dealing with complicated arithmetic involving decimals. The chain of simplifications is why polynomial integration, especially with substitutions, can make seemingly difficult problems much easier.

The constant of integration, represented as \( C \), is a crucial component of indefinite integrals. When integrating, especially with indefinite integrals, it is essential to account for \( C \) because integration can produce a whole family of functions that differ by a constant.

This means that the integral of any function \( f(x) \) over a domain is not just a single function, but rather an entire set of functions differing only by a constant. Hence, whenever you integrate and find an antiderivative, always add \( C \) at the end of the expression.

• For our exercise, after integrating and transforming back to the original variable \( t \), the expression \( \frac{(3t + 2)^{3.4}}{10.2} \) includes \( + C \) to represent all possible vertical shifts of the solution.
• In any practical calculation, if a particular boundary condition or initial condition is known, \( C \) can be determined precisely.

The ambiguity embraced by \( C \) in indefinite integrals aptly considers all potential solutions, making it an indispensable part of integration.