Integrals of Exponential Functions: Videos & Practice Problems

In calculus, understanding the relationship between derivatives and integrals is crucial, especially when dealing with exponential functions. The integral of a general exponential function, expressed as \( b^x \), can be derived by reversing the process used to find its derivative. When differentiating \( b^x \), the result is \( b^x \cdot \ln(b) \). Consequently, to find the integral of \( b^x \), we divide \( b^x \) by \( \ln(b) \). This leads us to the integral formula:

$$ \int b^x \, dx = \frac{b^x}{\ln(b)} + C $$

Here, \( C \) represents the constant of integration, and it is important to note that the base \( b \) must be greater than zero and not equal to one, as these conditions are necessary for the definition of an exponential function.

To illustrate this rule, consider the integral of \( 7^x \). Applying the formula, we find:

$$ \int 7^x \, dx = \frac{7^x}{\ln(7)} + C $$

Next, we can apply this rule to more complex integrals involving sums of exponential functions. For example, to evaluate the integral of \( 3 \cdot \left(\frac{1}{2}\right)^x + 8^x \), we can separate the integral into two parts using the sum rule:

$$ \int \left(3 \cdot \left(\frac{1}{2}\right)^x + 8^x\right) \, dx = \int 3 \cdot \left(\frac{1}{2}\right)^x \, dx + \int 8^x \, dx $$

Utilizing the constant multiple rule, we can factor out the constant 3 from the first integral:

$$ = 3 \int \left(\frac{1}{2}\right)^x \, dx + \int 8^x \, dx $$

Now, applying the integral rule for each term, we have:

$$ 3 \cdot \frac{\left(\frac{1}{2}\right)^x}{\ln\left(\frac{1}{2}\right)} + \frac{8^x}{\ln(8)} + C $$

Next, we can simplify the expression using properties of exponents and logarithms. The term \( \left(\frac{1}{2}\right)^x \) can be rewritten as \( \frac{1^x}{2^x} \), and the natural logarithm of a fraction can be expressed as:

$$ \ln\left(\frac{1}{2}\right) = \ln(1) - \ln(2) = -\ln(2) $$

Thus, the integral simplifies to:

$$ = -\frac{3}{2^x \cdot \ln(2)} + \frac{8^x}{\ln(8)} + C $$

In conclusion, mastering the integration of exponential functions allows for the application of various rules and properties, enhancing problem-solving skills in calculus. As you continue practicing these concepts, you'll gain confidence in handling more complex integrals involving exponential functions.

In calculus, understanding how to integrate exponential functions is crucial, particularly the function \( e^x \). The integral of \( e^x \) can be derived from the general rule for integrating exponential functions of the form \( b^x \). When integrating \( e^x \), we apply the rule by recognizing that the base \( b \) is \( e \). The integral can be expressed as:

\[\int e^x \, dx = \frac{e^x}{\ln(e)} + C\]

Since \( \ln(e) = 1 \), this simplifies to:

\[\int e^x \, dx = e^x + C\]

This result indicates that the integral of \( e^x \) is the same as the original function, which aligns with the property of derivatives for \( e^x \).

When dealing with integrals that include constants multiplied by \( e^x \), such as \( 5e^x \), the constant multiple rule allows us to factor out the constant. Thus, we have:

\[\int 5e^x \, dx = 5 \int e^x \, dx = 5e^x + C\]

For more complex integrals, such as \( \int (3x^4 - \pi e^x + 2) \, dx \), we can break this down into separate integrals using the sum and difference rules:

\[\int (3x^4 - \pi e^x + 2) \, dx = \int 3x^4 \, dx - \int \pi e^x \, dx + \int 2 \, dx\]

Each integral can be solved individually:

1. For \( \int 3x^4 \, dx \), applying the constant multiple rule and the power rule gives:

\[\int 3x^4 \, dx = 3 \cdot \frac{x^5}{5} = \frac{3}{5}x^5\]

2. For \( \int \pi e^x \, dx \), we factor out \( \pi \) and use the integral of \( e^x \):

\[\int \pi e^x \, dx = \pi e^x\]

3. For \( \int 2 \, dx \), we simply have:

\[\int 2 \, dx = 2x\]

Combining these results, the final answer for the integral becomes:

\[\int (3x^4 - \pi e^x + 2) \, dx = \frac{3}{5}x^5 - \pi e^x + 2x + C\]

This comprehensive approach to integration allows for the application of various rules, enhancing our ability to tackle more complex functions involving exponential terms. Continued practice with these rules will solidify understanding and proficiency in integration.