Question

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{e^{2 x}+2 e^{x}+1}{e^{x}} d x $$

Short answer

The indefinite integral of the function \(\int \frac{e^{2 x}+2 e^{x}+1}{e^{x}} d x\) is \(e^x + 2x - e^{-x} + C\) where \(C\) is the constant of integration.

Step-by-step solution

Divide By The Exponential

First divide the fraction for every term in the numerator. So the function will become: \(e^{x} + 2 + e^{-x}\)

Apply Basic Integration

The integral can be solved by applying the basic integration rules. The integral of \(e^x\) is just \(e^x\) , and the integral of a constant is that constant times \(x\). For \(e^{-x}\), the integral is \(-e^{-x}\). Apply the rules to each term.

Write The Final Answer

By adding together the results of the integrals for every term, the indefinite integral for the initial function is obtained.

Key concepts

Basic Integration Formulas

Understanding basic integration formulas is a key step in solving indefinite integrals. Think of integration as the reverse of differentiation. Instead of finding the slope, you're finding the area under the curve. Some of the simplest rules help you integrate standard functions.

• Linear Function: \( \int a \cdot dx = ax + C \) where \( a \) is a constant.
• Exponential Function: \( \int e^{ax} \cdot dx = \frac{1}{a} e^{ax} + C \).
• Constant Function: \( \int c \cdot dx = cx + C \) where \( c \) is a constant value.

These formulas, although basic, form the backbone to tackle more complex integrals.
In the given problem, dividing the expression first allows for easy application of these formulas. You get a sum of functions each matched with its respective rule for integration.

Exponential Functions Integration

Exponential functions, often involving \(e\), are unique but straightforward once you know the drill. The exponential function is special because its derivative (and integral) is proportional to the function itself.
To integrate functions like \(e^x\), you simply apply the rule:

• \( \int e^x \cdot dx = e^x + C \).

Negative Exponents

With negative exponents, the process is much alike but with an extra step:

• \( \int e^{-x} \cdot dx = -e^{-x} + C \).

You generally multiply the result by the reciprocal of the exponent. This is why you see a negative sign when integrating \(e^{-x}\).
In our original exercise, recognizing and using these properties simplifies our work significantly.

Integration Techniques

Mastering integration techniques includes more than learning formulas, especially when dealing with combinations of functions. A few strategies streamline the process:

• **Division**: Break down a complex fraction by dividing each term separately.
• **Sum Rule**: Integrate each part of a sum or difference individually. \( \int (f(x) + g(x)) \cdot dx = \int f(x) \cdot dx + \int g(x) \cdot dx \).

In our exercise, dividing before integrating ensured each component was manageable.

Simplifying Before Integrating

Often, you should rearrange or simplify the expression first. This makes it easier to identify which rules to apply. Once divided, integrating each part of the fraction separately becomes achievable using basic rules.
By following these techniques, even seemingly complex integrals reveal their straightforward nature. Integrating becomes an exciting puzzle where you know exactly which piece fits where.