Problem 27 Find the value of $k$ that mak... [FREE SOLUTION]

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Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

$Math Studyset 91影视 Explanations$ Math

14 Edition

Chapter 6: Problem 27

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int 2 e^{4 x-1} d x=k e^{4 x-1}+C$$

Short Answer

Expert verified

k = \frac{1}{2}\.

Step by step solution

Identify the function to be integrated

The given integral is $\int 2 e^{4 x-1} d x$.

Use substitution method

Let $u = 4x - 1$. Then, $du = 4dx$, or $dx = \frac{1}{4} du$.

Substitute in the integral

Substitute $u = 4x - 1$ and $dx = \frac{1}{4} du$ into the integral: $\int 2 e^{4 x-1} d x = \int 2 e^{u} \frac{1}{4} du$.

Simplify the integral

Factor out the constants from the integral: $\int 2 e^{4 x - 1} d x = \frac{1}{2} \int e^{u} du$.

Integrate with respect to u

Since the integral of $e^{u}$ with respect to $u$ is $e^{u}$, we have $\frac{1}{2} e^{u} + C$.

Substitute back the value of u

Replace $u$ with $4x - 1$ to get the antiderivative in terms of x. Therefore, $\frac{1}{2} e^{4 x - 1} + C$.

Determine the value of k

Compare this result with the given form $\text{k}\text{ }e^{4 x-1}+C$. We see that $\text{k} = \frac{1}{2}$.

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Key Concepts

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

Integration by Substitution

Integration by substitution is a method used to simplify complex integrals. This method involves substituting a part of the integral with a new variable to make the integral easier to solve. To understand this, let's look at a basic idea: change of variable.

In our example, we started with the integral $\int 2 e^{4 x-1} \, dx$.

Here are the steps we followed:

First, we identified a substitution that could simplify the integral. We let $\text{ }u = 4 x - 1$. This means the integral becomes in terms of $u$.
We also need to change the differential $dx$ to $du$. By differentiating $u$ with respect to $x$, we get \, $du = 4 dx$ or $dx = \frac{1}{4} \, du$.
We then substituted $u$ and $dx$ in the integral:$\int 2 e^{4 x-1} \, dx = \int 2 e^{u} \frac{1}{4} \, du$.
The next step was to simplify the integral by factoring out the constants, which gave us: ${\frac{1}{2}} \int e^{u} \, du$.

This simplification makes the integral easier to solve.

Exponential Functions

Exponential functions are functions that have constants raised to variable powers. They have the general form $f(x) = a^x$, where $a$ is a constant. A special case is when $a = e$, where $e$ is the base of the natural logarithm, approximately equal to 2.71828.

When integrating exponential functions like $e^{u}$, the integral is straightforward:

For $\int e^u \, du$, the result is simply $ e^u + C$. This is because the derivative of $e^u$ is $e^u$.
Hence, when we integrated ${\frac{1}{2}} \int e^{u} \, du$, we got ${\frac{1}{2} e^u + C}$.
Finally, we substituted back $u = 4x - 1$ to return to our original variable, giving us ${\frac{1}{2} e^{4 x - 1} + C}$.

Recognizing how exponential functions are integrated is crucial in solving many calculus problems effectively.

Definite Integrals

While our problem focused on an indefinite integral (which includes a constant of integration $C$), it's important to understand definite integrals too.

Definite integrals are those that have upper and lower bounds. They represent the area under the curve for a specific interval. For example:

$\text{ }\int_{a}^{b} f(x) \, dx$ means finding the integral of $f(x)$ from $a$ to $b$.

To compute a definite integral, we first find the indefinite integral (antiderivative) of $f(x)$.
Then, we evaluate this antiderivative at the upper limit $b$ and subtract its value at the lower limit $a$.
This process provides the net area under the curve between $x = a$ and $x = b$.

Understanding definite integrals is important for various applications, including finding areas, volumes, and solving real-world problems.

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91影视

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int 2 e^{4 x-1} d x=k e^{4 x-1}+C$$

Short Answer

Step by step solution

Identify the function to be integrated

Use substitution method

Substitute in the integral

Simplify the integral

Integrate with respect to u

Substitute back the value of u

Determine the value of k

Key Concepts

Integration by Substitution

Exponential Functions

Definite Integrals

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