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Chapter 4: Problem 38

Use a CAS to find an antiderivative, then verify the answer by computing a derivative. (a) \(\int \frac{x}{x^{4}+1} d x\) (b) \(\int 3 x \sin 2 x \, d x\) (c) \(\int \ln x \, d x\)

Short Answer

Expert verified
The antiderivatives of given functions are \( F(x)=\frac{1}{2} \ln |x^{2}+1|\), \( G(x)=\frac{-3}{2}x \cos(2x) + \frac{3}{4} \sin(2x) \), and \( H(x)=x(\ln(x) - 1) \). Verification by calculating derivatives confirmed the validity of these results.

Step by step solution

01

Find Antiderivatives

Using a CAS, find antiderivatives of the given functions. - \( F(x)=\int \frac{x}{x^{4}+1} dx = \frac{1}{2} \ln |x^{2}+1|\) - \( G(x)=\int 3x sin(2x) dx = \frac{-3}{2}x \cos(2x) + \frac{3}{4} \sin(2x) \) - \( H(x)=\int \ln(x) dx = x(\ln(x) - 1) \)
02

Verify the Result

Verify the result by calculating the derivatives of the obtained antiderivatives, which should match with the given functions. - \( F'(x)= (\frac{1}{2} \ln |x^{2}+1|)' = \frac{x}{x^{4}+1}\) - \( G'(x)= (\frac{-3}{2}x \cos(2x) + \frac{3}{4} \sin(2x))' = 3x \sin(2x) \) - \( H'(x)= (x(\ln(x) - 1))' = \ln(x) \)

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Key Concepts

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

Integral Calculus
Integral calculus is an essential branch of mathematics focused on the process of integration, which is the inverse operation to differentiation. At its heart lies the concept of the antiderivative, a function that reverses the action of taking a derivative. In simpler terms, if you have a function, integrating it allows you to find the original function of which it is the derivative.

In the given exercise, integral calculus is applied to find the antiderivatives of various functions. For example, the exercise \(a\) calculates the antiderivative of the function \(\frac{x}{x^{4}+1}\), which involves integrating it with respect to \(x\).
  • Understanding the power rule and the logarithm function is critical in solving these integrals.
  • Identifying appropriate substitutions or transformations, such as using partial fractions, if required.
  • Recognizing trigonometric integrals and their specific integral formulas, like those involving \(\sin\) and \(\cos\).
These elements are fundamental to solving integrals and require a strong foundation in the relationships between functions and their antiderivatives.
Derivative Verification
Derivative verification is a technique used to confirm the correctness of an antiderivative. In the context of calculus, once we've determined a potential antiderivative, we differentiate it and check if we obtain the original function.

This process serves as a proof that our initial integration was indeed correct, as differentiation is the inverse process of integration. For instance, in the provided solutions, the antiderivative \( F(x) \), when differentiated, should yield the original integrand \( \frac{x}{x^{4}+1} \).
  • Applying derivative rules, such as the chain rule, the product rule, and the quotient rule, allows us to differentiate complex functions efficiently.
  • Recognizing the derivatives of basic functions, including trigonometric, exponential, and logarithmic functions.
Mastery of these differentiation techniques ensures accuracy in derivative verification and solidifies our understanding of the integral calculus's fundamental theorem.
Computer Algebra System
A Computer Algebra System (CAS) is a sophisticated software tool that can perform symbolic mathematics. Students and professionals use CAS to solve complex mathematical problems, including integrals and derivatives, without tedious manual calculations.

In the exercise, a CAS is utilized to find antiderivatives of several functions, which might otherwise be difficult to integrate by hand. The integration capabilities of a CAS are based on built-in algorithms that can handle a wide range of functions and are especially helpful with

Non-Standard Integrals

Non-standard integrals are those that do not fit easily into basic integral forms and may involve complex manipulation or special functions.

Speed and Efficiency

Using a CAS massively reduces the time taken to perform integrations and allows for quick explorations of different approaches.

Error Reduction

Manual computations are prone to errors, especially with complicated integrals; a CAS helps in reducing such errors and improves accuracy.
CAS tools can be a great aid in learning integral calculus, allowing students to focus on understanding concepts rather than getting bogged down with intricate calculations.

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus to find an antiderivative of \(\sin \sqrt{x^{2}+1}\)

In most of the calculations that you have done, it is true that the Trapezoidal Rule and Midpoint Rule are on opposite sides of the exact integral (i.e., one is too large, the other too small). Also, you may have noticed that the Trapezoidal Rule tends to be about twice as far from the exact value as the Midpoint Rule.Given this, explain why the linear combination \(\frac{1}{3} T_{n}+\frac{2}{3} M_{n}\) should give a good estimate of the integral. (Here, \(T_{n}\) represents the Trapezoidal Rule approximation using \(n\) partitions and \(M_{n}\) the corresponding Midpoint Rule approximation.)

The data come from a pneumotachograph, which measures air flow through the throat (in liters per second). The integral of the air flow equals the volume of air exhaled. Estimate this volume. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 \\ \hline f(t)(1 / \mathrm{s}) & 0 & 0.1 & 0.4 & 0.8 & 1.4 & 1.8 & 2.0 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 \\ \hline f(t)(1 / \mathrm{s}) & 2.0 & 1.6 & 1.0 & 0.6 & 0.2 & 0 \\ \hline \end{array}$$

Let \(f\) be a continuous function on the interval \([0,1],\) and \(\mathrm{de}-\) fine \(g_{n}(x)=f\left(x^{n}\right)\) for \(n=1,2\) and so on. For a given \(x\) with \(0 \leq x \leq 1,\) find \(\lim _{n \rightarrow \infty} g_{n}(x) .\) Then, find \(\lim _{n \rightarrow \infty} \int_{0}^{1} g_{n}(x) d x\)

The sigmoid function \(f(x)=\frac{1}{1+e^{-x}}\) is used to model situations with a threshold. For example, in the brain, each neuron receives inputs from numerous other neurons and fires only after its total input crosses some threshold. Graph \(y=f(x)\) and find \(\lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow \infty} f(x) .\) Define the function \(g(x)\) to be the value of \(f(x)\) rounded off to the nearest integer. What value of \(x\) is the threshold for this function to switch from "off" (0) to "on" (1)? How could you modify the function to move the threshold to \(x=4\) instead?
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