Problem 26

Question

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

Step-by-Step Solution

Verified

Answer

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

1Step 1: 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.

2Step 2: 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} $.

3Step 3: Substitute and simplify

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

4Step 4: Integrate

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

5Step 5: Back-substitute for x

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

6Step 6: Obtain the final result

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

Key Concepts

Exponential Functionsu-substitutionAntiderivative

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.

Problem 26

Other exercises in this chapter

Problem 26

Evaluate. (Be sure to check by differentiating!) $$ \int t^{2}\left(t^{3}-1\right)^{7} d t $$

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Problem 26

Evaluate using integration by parts or substitution. (Assume $u>0$ in $\ln$ u. Check by differentiating. $$ \int x^{-5} \ln x d x $$

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Problem 26

Find the area of the region bounded by the graphs of the given equations. $$ y=x^{2}, y=x^{3} $$

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Problem 26

Find the area under the graph of each function over the given interval. $$ y=x^{4} ; \quad[0,1] $$

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