Problem 38 - Calculus and its applications

Question

Find each integral. $$ \int\left(2 x^{5}-4 e^{3 x}\right) d x $$

Step-by-Step Solution

Answer

The integral is \(\frac{x^6}{3} - \frac{4}{3}e^{3x} + C\).

Step 1: Break Down the Integral

The given integral is \(\int \left(2x^5 - 4e^{3x}\right) \, dx\). We can separate this into two integrals: \(\int 2x^5 \, dx - \int 4e^{3x} \, dx\). Our task is to solve these two integrals individually.

Step 2: Solve the Integral \(\int 2x^5 \, dx\)

Find the antiderivative of \(2x^5\). The antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). Thus, \(\int 2x^5 \, dx = 2 \cdot \frac{x^{6}}{6} = \frac{x^6}{3}\). Don't forget the constant of integration \(C_1\).

Step 3: Solve the Integral \(\int 4e^{3x} \, dx\)

Find the antiderivative of \(4e^{3x}\). The antiderivative of \(e^{ax}\) is \(\frac{e^{ax}}{a}\). Therefore, \(\int 4e^{3x} \, dx = 4 \cdot \frac{e^{3x}}{3} = \frac{4}{3}e^{3x}\). Don't forget the constant of integration \(C_2\).

Step 4: Combine the Solutions

Combine the antiderivatives found in Steps 2 and 3. This gives us the complete antiderivative: \(\frac{x^6}{3} - \frac{4}{3}e^{3x} + C\), where \(C\) is a constant that summarizes \(C_1 + C_2\).

Key Concepts

Antiderivative

In calculus, finding an antiderivative is about determining a function that reverses the process of differentiation. An antiderivative of a function exists if there is another function such that when you differentiate it, you obtain the original function. This is often referred to as an "integral" or "indefinite integral." The key property that distinguishes an antiderivative is the presence of a constant of integration, typically represented as "+ C," to account for any constant term that would vanish under differentiation. This is because when differentiating, constants disappear, so the antiderivative must include this arbitrary constant to cover all possible original functions.When tackling a problem needing an antiderivative, it's crucial to recognize the types of functions involved. For example, polynomial functions use the formula \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] In this way, understanding the rules of integration for different functions forms the basis of finding antiderivatives.

Exponential Function

Exponential functions are significant in calculus due to their unique property of maintaining their form when differentiated or integrated. The exponential function, typically in the form \( e^{ax} \), where \( e \) is Euler's number, is particularly interesting. When you integrate \( e^{ax} \), the process involves dividing by the constant \( a \) which appears in the exponent, resulting in \[ \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \]This simplicity in the integration process makes exponential functions straightforward to handle relative to other functions. They appear frequently in growth and decay problems and are vital in modeling continuous processes in nature and finance. By utilizing the rule for integration, as shown in the original exercise, we're able to systematically determine the antiderivative of any exponential function with a linear exponent.

Polynomial Integration

Integrating polynomials is a core part of calculus that students often practice to solidify their understanding of integral calculus. Polynomials consist of terms like \( ax^n \), where \( a \) is a constant, and \( n \) is a non-negative integer. To integrate a polynomial term like \( 2x^5 \), you apply the power rule of integration:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] In this specific exercise, taking the integral of \( 2x^5 \) provides \( \int 2x^5 \, dx = \frac{x^6}{3} + C \). It's crucial to always increase the exponent by one and divide by this new exponent. Each term of a polynomial is handled individually and then combined to find the entire antiderivative, making polynomial integration systematic and straightforward. Practice helps reinforce how constants and exponents interact during the integration process, leading to a deeper understanding of calculus fundamentals.