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Chapter 9: Problem 17

Integrate: $$\int e^{4 x} e^{x} d x$$

Short Answer

Expert verified
\(\frac{1}{5} e^{5x} + C\)

Step by step solution

01

Combine the Exponents

Use the property of exponents that states \[a^m \cdot a^n = a^{m+n}\]. Therefore, combine the exponents: \[e^{4x} \cdot e^x = e^{(4x + x)} = e^{5x}\]. Now the integral becomes \[ \int e^{5x} \, dx\].
02

Integrate the Expression

The integral of \[e^{ax} \, dx\] is \[\frac{1}{a} e^{ax} + C\], where \[a\] is a constant and \[C\] is the constant of integration. In this problem, \[a = 5\]. Therefore, we have: \[\int e^{5x} \, dx = \frac{1}{5} e^{5x} + C\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Exponents
Understanding the properties of exponents is crucial when dealing with exponential functions in integration. In the given exercise, we use one of the fundamental properties: \(a^m \times a^n = a^{m+n}\). This property allows us to combine exponential terms when the bases are the same.

For example, in \[e^{4x} \times e^x = e^{(4x + x)} = e^{5x}\], we start with two exponentials with the same base, \(e\). By adding the exponents (4x and x), we simplify the expression.

Mastering this property will help you simplify and solve more complex exponential integrals.
Integration of Exponential Functions
Integrating exponential functions is a common task in calculus. In our exercise, once we simplified the expression to \[e^{5x}\], we needed to integrate it.

The integral of a general exponential function \[e^{ax} \, dx\] is \(\frac{1}{a} e^{ax} + C\).

So in our case, we have: \[\int e^{5x} \, dx = \frac{1}{5} e^{5x} + C\], where \(a\) is 5.

Remember that integrating exponentials often involves multiplying by the reciprocal of the constant in the exponent. This technique helps us solve many integral problems involving exponentials more easily.
Constant of Integration
When performing indefinite integrals, it's essential to include the constant of integration, denoted by \(C\).

This constant represents an unknown value that could be any number since antiderivatives are unique up to a constant.

In the solution, after integrating \[e^{5x}\], we wrote: \[\frac{1}{5} e^{5x} + C\].

Never forget to add \(C\) when writing your final answer for indefinite integrals. Not including it can lead to incorrect solutions in further steps of the problem or in more complex applications.

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