Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-16, perform the indicated integrations. \(\int \frac{d x}{\left(x^{2}+4\right)^{3 / 2}}\)

Step-by-Step Solution

Answer

\( \frac{x}{4\sqrt{x^2 + 4}} + C \)

Step 1: Identify the Type of Integral

The given integral \( \int \frac{dx}{(x^2 + 4)^{3/2}} \) is a standard type of integral that involves a function of \( x^2 + a^2 \). This suggests the use of trigonometric substitution as a potential method for solving the integral.

Step 2: Choose the Appropriate Substitution

For integrals involving \( x^2 + a^2 \), the substitution \( x = a\tan(\theta) \) is useful. Here, choose \( x = 2\tan(\theta) \), since \( a = 2 \). Then, \( dx = 2\sec^2(\theta) d\theta \). Substitute these into the integral.

Step 3: Substitute and Simplify

Substituting \( x = 2\tan(\theta) \) and \( dx = 2\sec^2(\theta) d\theta \) into the integral, we have: \[\int \frac{2\sec^2(\theta) d\theta}{(4\tan^2(\theta) + 4)^{3/2}}\]Simplify the expression inside the integral. Notice that \( 4\tan^2(\theta) + 4 = 4(\tan^2(\theta) + 1) = 4\sec^2(\theta) \). Thus, the integral becomes:\[\int \frac{2\sec^2(\theta) d\theta}{(4\sec^2(\theta))^{3/2}}\]

Step 4: Simplify Further and Integrate

Simplify the denominator: \[(4\sec^2(\theta))^{3/2} = (2^2\sec^2(\theta))^{3/2} = 2^3 (\sec^2(\theta))^{3/2} = 8\sec^3(\theta)\]The integral simplifies to:\[\int \frac{2}{8\sec(\theta)} d\theta = \frac{1}{4} \int \cos(\theta) d\theta \]The integral of \( \cos(\theta) \) is \( \sin(\theta) \), so we have:\[\frac{1}{4} \sin(\theta) + C\]

Step 5: Back-Substitute for x

Recall the substitution \( x = 2\tan(\theta) \) which gives \( \tan(\theta) = \frac{x}{2} \). Thus, \( \sin(\theta) = \frac{x}{\sqrt{x^2 + 4}} \). Substitute \( \sin(\theta) \) back into the result:\[\frac{1}{4} \cdot \frac{x}{\sqrt{x^2 + 4}} + C\]This gives the antiderivative in terms of \( x \).

Step 6: Write the Final Answer

The final answer to the integral is:\[\int \frac{dx}{(x^2 + 4)^{3/2}} = \frac{x}{4\sqrt{x^2 + 4}} + C\]

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that focuses on finding the function that describes the accumulated area under a curve between certain points. When faced with an integral, we aim to determine a function whose derivative matches the integrand. This process is known as finding an antiderivative. The area or "integral" often helps in evaluating quantities like area, volume, and displacement.

• Integrals can be definite or indefinite: a definite integral provides a numerical answer, while an indefinite integral results in a general form, typically including a constant of integration (like + C).
• In the given problem, we work with an indefinite integral involving a trigonometric substitution, which requires us to transform the integrand to simplify the calculation.

Recognizing the structure of the integral, such as the form involving \(x^2 + a^2\), often suggests the technique to solve it, streamlining the process of finding the solution.

Antiderivative

The antiderivative is essentially the reverse of a derivative. If you can differentiate a function to get another, finding an antiderivative means you are identifying the original function from its derivative. Integral calculus revolves around these concepts. To solve the problem\( \int \frac{dx}{(x^2 + 4)^{3/2}} \), the goal is to find the original function whose derivative would result in the given integrand.

• The antiderivative includes an unknown constant represented by \( C \), because differentiation of constants equals zero, which means a constant could have been part of the original function.
• In our specific problem, by employing trigonometric substitution, we successfully determine a form for the antiderivative involving \( \sin(\theta) \) which we later convert back to \( x \).

This step-by-step process of finding the antiderivative is crucial for understanding and mastering integral calculus.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that are true for all values of the included variables. They are invaluable tools in simplifying complex integral computations, especially when substitution is involved.The key identity employed here is the Pythagorean identity: \[ \tan^2(\theta) + 1 = \sec^2(\theta) \]This identity allows us to transform the integrand into a form that is easier to integrate. Specifically, it simplifies expressions like\( 4\tan^2(\theta) + 4 \) to \( 4\sec^2(\theta) \), which enables further simplification of integrals. Mastery of these identities is crucial, as they often reveal a clear path to integration.

• Understanding these identities helps in transforming integral bounds from \( x \) to a trigonometric function, thus simplifying integration tasks.

For challenging integrals involving radicals, recognizing when to apply these identities is often the key to solving them efficiently.

Substitution Method

The substitution method is a powerful technique to simplify and solve integrals. When faced with complex integrands, the substitution method transforms the integral into a simpler form, often by changing the variable of integration. For functions of the type \( x^2 + a^2 \), like in our specific problem, the substitution \( x = a\tan(\theta) \) is typically effective.

• In our example, choosing \( x = 2\tan(\theta) \) helps convert the integral into a more manageable form using trigonometric identities.
• This change of variables can often turn a difficult integral into one involving basic trigonometric functions, such as \( \cos(\theta) \), which are simpler to integrate.

After integrating, it's essential to reverse the substitution to express the answer back in terms of the original variable \( x \). The substitution method not only makes integration feasible but also enhances the clarity and execution of solving integral calculus problems.