Problem 45 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{d x}{17+x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{1}{\sqrt{17}} \arctan\left(\frac{x}{\sqrt{17}}\right) + C\)

Step 1: Identify the Integral Form

The integral \(\int \frac{d x}{17+x^{2}}\) matches the standard form of \(\int \frac{d x}{a^2 + x^2}\), which is known to result in \(\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\) where \(C\) is the constant of integration.

Step 2: Determine the Parameter "a"

In the given integral, compare it with the standard form. Here, \(a^2 = 17\), therefore \(a = \sqrt{17}\).

Step 3: Apply the Integral Formula

Now, substitute \(a = \sqrt{17}\) into the formula for the standard integral. This gives us \(\frac{1}{\sqrt{17}} \arctan\left(\frac{x}{\sqrt{17}}\right) + C\).

Step 4: Write the Final Answer

Therefore, the evaluated integral is \(\frac{1}{\sqrt{17}} \arctan\left(\frac{x}{\sqrt{17}}\right) + C\).

Key Concepts

Integration Techniques

Integration techniques are methods used to find the antiderivative or integral of a function. One useful technique involves recognizing standard forms of integrals.

• By matching an integral to a known form, we can directly apply a formula to solve it.
• This saves time and reduces complexity, making integration more approachable.

For example, the integral \( \int \frac{d x}{a^2 + x^2} \) is a classic form, solved using a straightforward formula involving arctangent. Recognizing such patterns is key in simplifying the process.
Different problems may require different techniques. Common methods include substitution, integration by parts, and partial fraction decomposition. Each technique has its unique application scenarios for greater efficiency. Efficiency in integration depends largely on selecting the right technique for the problem at hand.

Definite and Indefinite Integrals

Integrals can be classified as definite and indefinite integrals, each serving a different purpose and involving distinct concepts:

• An indefinite integral, like \( \int f(x) \, dx \), represents a family of functions whose derivative gives the original function \( f(x) \).
• It includes a constant of integration, usually denoted by \( C \).
• Indefinite integrals capture the general form of antiderivatives.

On the other hand, a definite integral evaluates the area under the curve of \( f(x) \) from one point to another.

• It is represented as \( \int_{a}^{b} f(x) \, dx \).
• Unlike indefinite integrals, definite integrals result in a specific number without the constant \( C \).
• They are very useful in calculating total quantities, such as distances or areas between curves.

Understanding the differences between these two types of integrals helps in applying them correctly in various situations. Picking the correct type ensures clear and useful conclusions in solving mathematical problems.

Trigonometric Substitution

Trigonometric substitution is a powerful integration technique used to simplify integrals involving expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), and \( \sqrt{x^2 - a^2} \).

• It uses trigonometric identities to transform the integrals into more manageable forms.
• By substituting variables with trigonometric functions, complex algebraic forms become simpler trigonometric expressions.

Here's how it works:
Consider the integral \( \int \frac{d x}{a^2 + x^2} \).

• Recognizing it as an inverse tangent form allows for easier calculation.
• For similar expressions, substitution like \( x = a \tan(\theta) \) helps convert it to a trigonometric integral.
• These substitutions leverage angles and trigonometric function properties to simplify further steps.

Trigonometric substitution is particularly useful when dealing with integrals derived from circular and hyperbolic functions. Mastering these techniques ensures more effective and simplified solutions.