Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int\left(1+x+e^{x}\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral of the given function is: $$\int (1+x+e^x) dx = x + \frac{x^2}{2} + e^x + C$$

Step 1: Integrate the constant term

For the constant term, we have: $$\int 1 dx$$ The indefinite integral of a constant is just the constant itself multiplied by the independent variable. So, we have: $$\int 1 dx = x + C_1$$ Where \(C_1\) is the constant of integration.

Step 2: Integrate the linear term

For the linear term, we have: $$\int x dx$$ Use the power rule, which states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) for any real number \(n\neq-1\), to find the indefinite integral: $$\int x dx = \frac{x^{1+1}}{1+1} + C_2 = \frac{x^2}{2} + C_2$$ Where \(C_2\) is another arbitrary constant of integration.

Step 3: Integrate the exponential term

For the exponential term, we have: $$\int e^x dx$$ The indefinite integral of an exponential function (with base \(e\)) is the same function multiplied by a constant factor. In this case, we have: $$\int e^x dx = e^x + C_3$$ Where \(C_3\) is another constant of integration.

Step 4: Combine the integrals

Now, we will add the results of the three integrals together to find the indefinite integral of the original function: $$\int (1+x+e^x) dx = \left(x + C_1\right) + \left(\frac{x^2}{2} + C_2\right) + \left(e^x + C_3\right)$$ Combining the constants of integration \(C_1, C_2\), and \(C_3\) into a single constant \(C\), we get the final result: $$\int (1+x+e^x) dx = x + \frac{x^2}{2} + e^x + C$$

Key Concepts

Integration Techniques

Integration is an essential mathematical tool used to find the area under a curve, reverse the process of differentiation, or solve various mathematical problems involving rates of change and accumulation. There are several integration techniques available, each suited for different types of functions. When facing an integral like \(\int(1+x+e^{x}) dx\), it's helpful to recognize that each term requires a different method.

For simpler integrals involving constants or basic power functions, we rely on the fundamental formulas of integration. More complex integrals may require techniques such as substitution, integration by parts, partial fractions, or trigonometric integration. In our example, we integrate each term separately using basic integration rules, which are sufficient for the given terms.

Power Rule of Integration

The power rule of integration is one of the most basic and frequently used techniques for integrating polynomials. It states that to integrate a term like \(x^n\), where \(n\) is a real number except -1, we increase the power by 1 and divide by the new power, adding a constant of integration at the end. Following this rule, for the integral \(\int x dx\), we apply \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) to obtain \(\frac{x^2}{2} + C\).

In our example, using the power rule simplifies the process significantly and makes the integration of polynomial terms straightforward. Remembering and applying the power rule is crucial for students to solve integrals efficiently.

Exponential Function Integration

When dealing with the exponential function \(e^x\), its indefinite integral is uniquely straightforward. The integral of \(e^x\) with respect to \(x\) is simply \(e^x + C\), where \(C\) is the constant of integration. This property arises from the fact that the rate of change of the function \(e^x\) with respect to \(x\) is the function itself.

In our textbook example, for the term \(\int e^x dx\), we apply this rule directly and swiftly obtain \(e^x + C\) as the solution. Understanding the simplicity of exponential functions and their integrals can save students a lot of time and avoid unnecessary complexity when encountering such terms in an integral.