Problem 40 - Calculus: One and Several Variables

Question

Calculate the integral: (a) by integrating by parts, (b) by applying a trigonometric substitution. $$\int x \arctan x d x$$

Step-by-Step Solution

Answer

The integral of \(x \arctan x\) is given by: \(\int x \arctan x d x = x\arctan x - \frac{x^2}{2} + C\)

Step 1: Select u and dv

Let's select: \(u = x\) and \(dv = \arctan x d x\) Now we differentiate \(u\) and integrate \(dv\): \(du = d x\) and \(v = \int \arctan x d x\)

Step 2: Apply Integration by Parts Formula

Now we apply the integration by parts formula: \(\int x \arctan x d x = x \int \arctan x d x - \int (\int \arctan x d x) d x\) This gives us: \(\int x \arctan x d x = x \int \arctan x d x - \int v d x\) Now we need to find \(v\) by integrating \(\arctan x\), which is a bit complex, and this may not be the most suitable approach for this specific integral. (b) Trigonometric Substitution

Step 1: Choose Trigonometric Substitution

We choose an appropriate trigonometric substitution. Let: \(x = \tan \theta\), where \(-\frac{\pi}{2}< \theta <\frac{\pi}{2}\) This gives us \(d x = \sec^2 \theta d \theta\)

Step 2: Rewrite the Integral with Substitution

Substituting the expressions for \(x\) and \(dx\) in the given integral, we get \(\int x \arctan x d x = \int \tan \theta \arctan (\tan \theta) \sec^2 \theta d \theta\)

Step 3: Apply Simplification

Now, we simplify the expression inside the integral: \(\int \tan \theta \arctan (\tan \theta) \sec^2 \theta d \theta= \int \theta \cdot \sec^2 \theta d \theta\)

Step 4: Integration

Now, we integrate w.r.t \(\theta\): \(\int \theta \cdot \sec^2 \theta d \theta = \theta\int \sec^2 \theta d\theta - \int [\int \sec^2 \theta d\theta] d\theta\) By calculating the integral, we get: \(\int x \arctan x d x= \theta \tan \theta - \int \tan \theta d\theta \) Now we reverse the substitution to express the result in terms of x: \(\int x \arctan x d x= \arctan x \cdot x - \int x d x\)

Step 5: Final Integration and Result

Finally, we integrate the remaining term and obtain the result: \(\int x \arctan x d x = \arctan x \cdot x - \frac{x^2}{2} + C\) So, the integral of \(x \arctan x\) is given by: \(\int x \arctan x d x = x\arctan x - \frac{x^2}{2} + C\)

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to solve integrals involving square roots, quadratic expressions, and certain other non-polynomial functions. When direct integration is difficult or impossible, this method can simplify the integral into a form that is more manageable. The basic strategy involves replacing a variable with a trigonometric function of a new variable, typically \theta, which often turns out to be a more solvable expression.

For example, in the integral \(\int x \arctan x dx\), a trigonometric substitution of \(x = \tan \theta\) transforms the integral into a new form involving \(\theta\) and trigonometric functions. This substitution makes the integral easier because \(\arctan(\tan \theta) = \theta\) simplifies the expression, leaving behind a new integral that can be directly integrated with respect to \(\theta\).

After integration, we must not forget to convert the result back to the original variable using the inverse trigonometric function. This step is crucial to express the solution in terms of the original variable.

Indefinite Integral

The indefinite integral, also known as antiderivative, is essentially the reverse process of differentiation. When we find the indefinite integral of a function, we are looking for a function whose derivative is the given function. A key characteristic of indefinite integrals is that they include an arbitrary constant, often denoted as \(C\), because the derivative of a constant is zero. This means there are an infinite number of functions that could be the antiderivative of a given function, each differing by a constant.

The notation for an indefinite integral is \(\int f(x) dx\) and finding this integral involves determining the function \(F(x)\) such that \(\frac{d}{dx}F(x) = f(x)\). Integrals can represent a variety of concepts such as area, work, or cumulative totals. In the exercise \(\int x \arctan x dx\), we are interested in finding the function whose derivative gives us \(x \arctan x\), and we add a constant \(C\) to our final answer.

Integration Techniques

Various integration techniques are used to solve different types of integrals, some of which may be complex and not directly solvable using basic methods. Among these techniques are integration by parts, trigonometric substitution, partial fractions, and u-substitution. Knowing when and how to apply these techniques is crucial for solving integrals effectively.

Integration by parts, for instance, is based on the product rule of differentiation and is used when an integral is the product of two functions. It involves choosing one function to differentiate (\(u\)) and another to integrate (\(dv\)) and applying the formula \(\int u dv = uv - \int v du\). This is a strategic choice and requires some practice to determine the best choice for \(u\) and \(dv\) that will simplify the integral.

Understanding these integration techniques and practicing their application will help to solve a broader range of integrals more efficiently and is a fundamental skill in calculus. Each technique has its own set of rules and scenarios in which it is most useful, making it essential to develop a strong grounding in each method.