Problem 38 - Calculus: Early Transcendental Functions

Question

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) \(\int \ln x \, d x\)

Step-by-Step Solution

Answer

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 1: Find Antiderivatives

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

Step 2: Verify the Result

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

Key Concepts

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.