91Ó°ÊÓ

Integrals can be broadly classified into two types: definite and indefinite. An indefinite integral is an anti-derivative function, essentially representing a family of functions. It is denoted without upper or lower limits and includes an arbitrary constant (denoted by C). For example, \(\frac{1}{7} e^{7x} + C\) is an indefinite integral.

Alternatively, a definite integral computes the area under the curve of a function between two specified bounds, providing a scalar value. The notation includes upper and lower limits, like so: \[\int_{a}^{b} f(x) \ dx\].

When solving integrals, make sure to understand if you are dealing with a definite or an indefinite integral as it affects the method and the final answer.

The substitution method in integration is a popular technique to simplify integrals. It involves substituting part of the integral with a new variable, making the integral easier to evaluate.

In our example, we have the integral \(\int e^{7x} \ dx\).

Here's how to apply the substitution method step by step:

• Identify a part of the integral to substitute. Here, let \(u = 7x\).
• Differentiate the substitution to find \(du\), which gives \(du = 7 \ dx\), and solve for \(dx = \frac{du}{7}\).
• Substitute both \(u\) and \(dx\) back into the original integral: \[ \int e^{7x} \ dx = \int e^u \frac{du}{7} = \frac{1}{7}\ \int e^u \ du.\]
• Integrate \(e^u\), yielding \(\frac{1}{7} e^u + C\).
• Substitute back \(u = 7x\) to return to the original variable: \(\frac{1}{7} e^{7x} + C\).

Using substitution simplifies complex integrals significantly. Just remember to replace all occurrences, including \(dx\), carefully.

One essential step after solving an integral is to verify your solution through differentiation. This ensures your anti-derivative function correctly translates back to the original function.

Let's verify our integral \(\frac{1}{7} e^{7x} + C\).

• Differentiate \(\frac{1}{7} e^{7x} + C\):
  \(\frac{d}{dx} \left( \frac{1}{7} e^{7x} + C\right) = \frac{1}{7} \cdot 7 e^{7x} = e^{7x}\).
  The differentiation matches the original integrand \(e^{7x}\), confirming our solution.

This step is crucial as it validates that the integral was solved correctly, reinforcing your understanding and ensuring accuracy in your work.