Understanding the Integral of the Form $\int e^x (f(x) + f'(x)) \, dx$

Integrals of the form $\int e^x (f(x) + f'(x)) \, dx$ are interesting because they can often be solved using a simple and elegant technique known as integration by parts or by recognizing a pattern that simplifies the integral. Let's delve deeper into this topic to understand how to approach such integrals, especially in the context of preparing for exams.

Integration by Parts

Integration by parts is a technique that comes from the product rule for differentiation and is used to integrate products of functions. The formula for integration by parts is given by:

[ \int u \, dv = uv - \int v \, du ]

where $u$ and $dv$ are differentiable functions of $x$. To apply this to our integral, we need to identify parts of the integrand as $u$ and $dv$.

Recognizing a Pattern

Sometimes, the integral $\int e^x (f(x) + f'(x)) \, dx$ can be simplified by recognizing that it is the derivative of a product of functions. Notice that if we have a function $g(x) = e^x f(x)$, then its derivative is $g'(x) = e^x f(x) + e^x f'(x)$. This observation leads to a direct integration:

[ \int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C ]

where $C$ is the constant of integration.

Table of Differences and Important Points

Feature	Integration by Parts	Direct Pattern Recognition
Technique	Requires identifying $u$ and $dv$	Recognizes the integrand as a derivative of a product
Formula	$\int u \, dv = uv - \int v \, du$	$\int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C$
Application	More general, works for various integrands	Specific to integrands of the form $e^x (f(x) + f'(x))$
Complexity	Can be more complex, may require additional steps	Very simple and straightforward

Examples

Let's look at some examples to illustrate these points.

Example 1: Direct Pattern Recognition

Consider the integral $\int e^x (2x + 2) \, dx$. Here, we can see that $f(x) = 2x$ and $f'(x) = 2$. Recognizing the pattern, we can directly integrate:

[ \int e^x (2x + 2) \, dx = e^x (2x) + C = 2xe^x + C ]

Example 2: Integration by Parts

Now, let's take an integral that doesn't fit the pattern perfectly: $\int x e^x \, dx$. We can use integration by parts here by choosing $u = x$ and $dv = e^x \, dx$. Then, $du = dx$ and $v = e^x$. Applying the formula:

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

In this case, integration by parts was necessary because the integrand was not of the form $e^x (f(x) + f'(x))$.

Conclusion

When faced with an integral of the form $\int e^x (f(x) + f'(x)) \, dx$, it is often most efficient to recognize the pattern that the integrand is the derivative of $e^x f(x)$. This allows for direct integration without the need for more complex techniques. However, when the integrand does not fit this pattern, integration by parts is a powerful tool that can be applied to a wide range of problems. Understanding both methods and when to apply them is crucial for solving integrals efficiently, especially in an exam setting.