Problem 26 Find each integral. $$ \int ... [FREE SOLUTION]

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Calculus and its applications

Bittinger, Marvin L. Ellenbogen, David J. Surgent

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 4: Problem 26

Find each integral. $$ \int 4 e^{4 x} d x $$

Short Answer

Expert verified

The integral is $ e^{4x} + C $.

Step by step solution

Identify the integral form

We recognize that the integral given is $ \int 4 e^{4x} \, dx $. This integral is in the form of $ \int e^{u} \, du $, where the integral of $ e^{u} $ is $ e^{u} + C $, where $ C $ is the constant of integration.

Perform u-substitution

Set $ u = 4x $ so that when you differentiate, $ \frac{du}{dx} = 4 $. This implies that $ du = 4 \, dx $. Solving for $ dx $ gives us $ dx = \frac{du}{4} $.

Substitute and simplify

Substitute $ u = 4x $ into the integral and replace $ dx $:\[ \int 4 e^{4x} \, dx = \int 4 e^{u} \left( \frac{du}{4} \right) = \int e^{u} \, du \].

Integrate

The integral $ \int e^{u} \, du $ is simply $ e^{u} + C $.

Back-substitute for x

Replace $ u $ with $ 4x $ to express the antiderivative in terms of $ x $: \[ e^{u} + C = e^{4x} + C \].

Obtain the final result

Thus, the original integral is: \[ \int 4 e^{4x} \, dx = e^{4x} + C \].

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Key Concepts

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

Exponential Functions

Exponential functions are a crucial concept in calculus and have the form $ f(x) = a e^{bx} $, where $ a $ is a constant, $ e $ is the base of the natural logarithm, approximately equal to 2.718, and $ b $ is the rate of growth or decay.
Exponential growth means the function increases rapidly as $ x $ increases, and exponential decay means it decreases rapidly as $ x $ increases.
Understanding exponential functions is essential because they model real-life situations like population growth, radioactive decay, and interest calculations.

u-substitution

U-substitution is a fundamental technique used in integration to simplify complex integrals.
The core idea is to make a substitution of a part of the integrand with a single variable $ u $, transforming the integral into a simpler form.

First, identify a part of the integrand function to replace with $ u $, usually where the derivative of $ u $ is present elsewhere in the integrand.
Differentiate $ u \, \text{with respect to} \, x $ to find $ \frac{du}{dx} $, and solve for $ dx $ in terms of $ du $.
Substitute both $ u $ and $ dx $ in the integral, simplifying it for easier integration.

This technique is especially useful when dealing with composite functions, as it simplifies the integration process significantly.

Antiderivative

An antiderivative of a function is a function whose derivative is the original function.
For example, the antiderivative of $ 4e^{4x} $ is a function $ F(x) $ such that $ F'(x) = 4e^{4x} $.

The process of finding an antiderivative is called indefinite integration, which leads to a family of functions differing by a constant $ C $.
The concept of an antiderivative is essential in solving differential equations, which describe many physical phenomena.
Remember, the notation $ \int f(x) \, dx $ represents the antiderivative of $ f(x) $.

Understanding how to find antiderivatives is essential because it is the reverse operation of differentiation.

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91影视

Find each integral. $$ \int 4 e^{4 x} d x $$

Short Answer

Step by step solution

Identify the integral form

Perform u-substitution

Substitute and simplify

Integrate

Back-substitute for x

Obtain the final result

Key Concepts

Exponential Functions

u-substitution

Antiderivative

One App. One Place for Learning.

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