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Chapter 4: Problem 14

1-20 Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(x)=3 e^{x}+7 \sec ^{2} x\)

Short Answer

Expert verified
Antiderivative: \( F(x) = 3e^x + 7\tan x + C \).

Step by step solution

01

Understanding the Function

We need to find the most general antiderivative of the function \( f(x) = 3e^x + 7\sec^2 x \). The general antiderivative involves finding the indefinite integral of each term of the function.
02

Antiderivative of the Exponential Term

The first term of the function is \( 3e^x \). The antiderivative of \( e^x \) is itself, \( e^x \). So, the antiderivative of \( 3e^x \) is \( 3e^x + C_1 \), where \( C_1 \) is a constant of integration.
03

Antiderivative of the Trigonometric Term

The second term of the function is \( 7\sec^2 x \). The antiderivative of \( \sec^2 x \) is \( \tan x \). Thus, the antiderivative of \( 7\sec^2 x \) is \( 7\tan x + C_2 \), where \( C_2 \) is another constant of integration.
04

Combine the Antiderivatives

Combine the antiderivatives found in Steps 2 and 3. The most general antiderivative of \( f(x) = 3e^x + 7\sec^2 x \) is \( F(x) = 3e^x + 7\tan x + C \), where \( C \) is the constant of integration (\( C = C_1 + C_2 \)).
05

Verify by Differentiation

Differentiate the antiderivative \( F(x) = 3e^x + 7\tan x + C \). The derivative of \( 3e^x \) is \( 3e^x \), the derivative of \( 7\tan x \) is \( 7\sec^2 x \), and the derivative of the constant \( C \) is 0. Thus, the derivative is \( 3e^x + 7\sec^2 x \), which matches the original function \( f(x) \).

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Key Concepts

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

Integration
Integration is like the opposite of differentiation. When you differentiate, you find the rate of change of a function. But when you integrate, you find the original function based on its rate of change.
For example, if you're given the function's derivative and need the original function, you perform integration. This process gives you the antiderivative, also known as the indefinite integral.
  • The symbol for integration is \( \int \).
  • When you integrate, you usually add a constant of integration, often represented as \( C \).
  • These constants account for the fact that there are multiple functions, differing by a constant, whose derivatives are the same.
In our exercise, we worked with the integration of functions like \( e^x \) and \( \sec^2 x \), realizing the antiderivative integrates the rate of change back to the original function.
Exponential Function
The exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It's one of the most important functions in calculus.
  • An example of an exponential function is \( e^x \), where \( e \) is approximately 2.71828.
  • Remarkably, \( e^x \) is its own derivative and its own antiderivative!
This unique characteristic makes it simpler to differentiate and integrate. In our example, the antiderivative of \( 3e^x \) is simply \( 3e^x + C \), as taking its derivative returns the original function. This property contributes to its frequent appearance in problems across math and science.
Trigonometric Function
Trigonometric functions, like sine, cosine, and tangent, are all about the angles and sides of triangles. They are crucial in calculus as they frequently appear in integrals and derivatives.
  • The function \( \sec^2 x \) is the derivative of \( \tan x \).
  • Therefore, when integrating \( \sec^2 x \), the antiderivative is \( \tan x \).
In our problem, the term \( 7\sec^2 x \) is integrated to become \( 7\tan x + C \). This illustrates how knowing the derivatives of trigonometric identities helps in easily finding their integrals. Understanding these identities can significantly enhance skills in solving integrals involving trigonometric functions.

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Most popular questions from this chapter

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f "\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x)=x^{2}-4 x+7 \cos x, \quad-4 \leqslant x \leqslant 4\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f "\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x)=\frac{x^{2}-1}{40 x^{3}+x+1}\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x)=c x+\sin x\)

At which points on the curve \(y = 1 + 40 x ^ { 3 } - 3 x ^ { 5 }\) does the tangent line have the largest slope?

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