Problem 20 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(6 e^{3 x}+4 x\right) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \( 2e^{3x} + 2x^2 + C \).

Step 1: Break Down the Integral

The integral given is \( \int \left(6e^{3x} + 4x\right) dx \). This is a sum of two separate integrals: \( \int 6e^{3x} \: dx \) and \( \int 4x \: dx \). We can tackle them individually.

Step 2: Integrate the Exponential Component

To integrate \( \int 6e^{3x} \: dx \), recognize it as a basic exponential integral form \( \int ae^{bx} \: dx = \frac{a}{b}e^{bx} + C \). Thus, \( \int 6e^{3x} \: dx = \frac{6}{3} e^{3x} = 2e^{3x} \).

Step 3: Integrate the Polynomial Component

For \( \int 4x \: dx \), use the power rule for integration, which states \( \int x^n \: dx = \frac{x^{n+1}}{n+1} + C \). Therefore, \( \int 4x \: dx = 4 \cdot \frac{x^2}{2} = 2x^2 \).

Step 4: Combine the Results

Now, combine the results from the integrated components: \( 2e^{3x} + 2x^2 \). Because we combined two indefinite integrals, we add an integration constant \( C \). So the final answer is \( 2e^{3x} + 2x^2 + C \).

Key Concepts

Understanding Integration Techniques

Integration techniques are mathematical methods used to find the antiderivative of a function. One common approach is to break down a complex integral into simpler parts, which can be handled individually. When given an integral like \(\int(6e^{3x} + 4x) \, dx\), the first step is to separate it into distinct integrals: \(\int 6e^{3x} \, dx\) and \(\int 4x \, dx\). This is because the integral of a sum is equal to the sum of the integrals.

By breaking them down, you apply different techniques suitable for each part. For polynomial functions, the power rule is often used, while for exponential functions involving \(e^{x}\), a special form is utilized. Understanding these techniques equips you to handle a wide range of calculus problems efficiently.

Exploring Exponential Functions in Integration

Exponential functions have a base of \(e\), the natural exponential constant approximately equal to 2.718. They are unique because their rate of growth is proportional to their value. This property makes them particularly interesting in integration. The basic formula for integrating exponential functions of the form \(ae^{bx}\) is \(\int ae^{bx} \, dx = \frac{a}{b}e^{bx} + C\).

In the exercise, we applied this formula to solve \(\int 6e^{3x} \, dx\). By identifying \(a = 6\) and \(b = 3\), we calculated it as \(\frac{6}{3}e^{3x} = 2e^{3x}\). This highlights how changes in the exponent's coefficient (the '3' in '3x') determine the integral's divider (the '\(b\)'), ensuring a correct antiderivation.

Integrating exponential functions is critical in many scientific fields due to their natural appearance in growth and decay processes.

Polynomial Functions and Their Integration

Polynomial functions are expressions involving variables raised to various powers and possibly multiplied by coefficients. For instance, a component of our exercise was \(4x\), which is a first-degree polynomial. The power rule is the key to integrating such functions, stating \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).

When faced with \(\int 4x \, dx\), apply the power rule: You increase the exponent of \(x\) by 1 (from \(1\) to \(2\)), and divide by this new exponent. This gives \(4 \cdot \frac{x^2}{2} = 2x^2\). Recognizing the pattern of adding 1 to the exponent and dividing by it can help solve polynomial integrals quickly.

Mastering the integration of polynomial functions is essential, as they form the foundation of many mathematical expressions encountered in calculus.