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Chapter 4: Problem 29

Determine these indefinite integrals. $$\int e^{7 x} d x$$

Short Answer

Expert verified
\(\frac{e^{7x}}{7} + C\)

Step by step solution

01

Recall the formula for integrating exponential functions

When integrating an exponential function of the form \(\int e^{ax} dx\), the result is \(\frac{e^{ax}}{a}\) plus a constant of integration (C).
02

Identify the values of a

In the given integral \(\int e^{7x} dx\), the value of \(a\) is 7.
03

Apply the integral formula

Substitute \(a = 7\) into the integral formula: \(\int e^{7x} dx = \frac{e^{7x}}{7} + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function Integration
Integrating exponential functions is a common task in calculus. Here, we'll look into how to tackle this type of integral.
Remember the general form of an exponential function is often written as \(\textstyle{e^{ax}}\). The key point in the integration process involves recognizing the 'a'.
When performing the integration of \(e^{ax}\), we utilize the following formula: \[\textstyle{\text{鈭玡^{ax} dx = \frac{e^{ax}}{a} + C}}\].
This formula applies to any exponential function where the exponent is a linear function of x.
In our example, \(a = 7\), thus we substitute 7 into the formula:

\[\textstyle{\text{鈭玡^{7x} dx = \frac{e^{7x}}{7} + C}}\].
Constant of Integration
When solving indefinite integrals, it's important to always include the constant of integration, denoted as C.
Since indefinite integrals represent a family of functions, there are infinitely many antiderivatives differing by a constant.
This means if you differentiate any antiderivative plus any constant, you will get back to the original function.
For example, given \(f(x) = \frac{e^{7x}}{7} + 5\) and \(g(x) = \frac{e^{7x}}{7} - 3\), both of these are valid antiderivatives of \(e^{7x}\).

Whenever you compute an indefinite integral, don't forget to represent this infinite family by adding + C to your result.
Integration Formula
Formulas help streamline calculus operations. For exponential functions, this is particularly true.
For integration, one handy formula to remember is the exponential integration formula:
\[\textstyle{\text{鈭玡^{ax} dx = \frac{e^{ax}}{a} + C}}\].
Using the correct formula ensures you apply consistent rules.
When you integrate \(e^{ax}\), recall each component:
  • 'a' in the exponent is crucial because it affects the denominator of the result.
  • Divide by 'a' to get the correct integral.
  • Always include the constant of integration, C.
In practice, breaking down the process ensures accuracy.
So, first identify a, apply the division, and remember to add C.

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