Problem 47 - Essential Calculus

Question

etermine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x=\tan \theta,\) then \(\int_{0}^{\sqrt{3}} \frac{d x}{\left(1+x^{2}\right)^{3 / 2}}=\int_{0}^{4 \pi / 3} \cos \theta d \theta .\)

Step-by-Step Solution

Answer

The statement is false. The two integrals result in different quantities, \(\frac{\sqrt{3}}{2}\) and \(-\frac{\sqrt{3}}{2}\). Thus they are not equivalent.

Step 1: Transform the variable

Transform the variable of the first integral using \(x=\tan \theta\). Thus, \(dx = \sec^2(\theta) d\theta\). Therefore, the first integral becomes \(\int \frac{\sec^2(\theta)}{(1+\tan^2(\theta))^{3/2}} d \theta\). We know from trigonometric identity that \(1+\tan^2 \theta = \sec^2 \theta\), so the integral simplifies to \(\int \frac{\sec^2(\theta)}{(\sec^2(\theta))^{3/2}} d \theta = \int \frac{d \theta}{\sec \theta} = \int \cos(\theta) d \theta \)

Step 2: Determine the limits of integration for \(\theta\)

Given \(x=\tan(\theta)\), we can determine the corresponding \(\theta\) values for \(x=0\) and \(x=\sqrt{3}\). These correspond to \(\theta=0\) and \(\theta=\frac{\pi}{3}\) respectively.

Step 3: Evaluate the transformed integral

Evaluate the integral of step 1 from \(\theta=0\) to \(\theta=\frac{\pi}{3}\). This gives \( \int_{0}^{\pi/3} \cos(\theta) d \theta = \sin(\theta) \bigg|_{0}^{\pi/3} = \frac{\sqrt{3}}{2} \)

Step 4: Evaluate the provided integral

Evaluate \(\int_{0}^{4\pi/3} \cos(\theta) d \theta \), it results to \(\sin(\theta) \bigg|_{0}^{4\pi/3} = -\frac{\sqrt{3}}{2} \)

Step 5: Compare the results obtained from Steps 3 and 4

We have the first integral equals \(\frac{\sqrt{3}}{2}\) from Step 3, and the second integral equals \(-\frac{\sqrt{3}}{2}\) from Step 4. They are not equal, therefore the original statement is false

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify certain integrals, especially those involving square roots of quadratic expressions. When the integrand involves expressions like \(1 + x^2\), substituting \(x = \tan \theta\) allows us to leverage the identity \(1 + \tan^2 \theta = \sec^2 \theta\). This substitution transforms the integrand into a trigonometric function, making the integral easier to evaluate.

The process of trigonometric substitution often involves replacing a variable with a trigonometric function, such as sine, cosine, or tangent. In our example, setting \(x = \tan \theta\) was crucial. This change of variables also transforms \(dx\) to \(\sec^2(\theta) d\theta\), altering the integral's form.

• Helps in solving integrals involving square roots
• Simplifies expressions using trigonometric identities
• Requires understanding of inverse trigonometric functions for reversing substitution

Once substitution is complete and the integral is evaluated, it is typically necessary to back-substitute to return to the original variable of the problem.

Integration Techniques

Integration techniques are methods applied to solve integrals, especially when straightforward antiderivatives are not evident. In this exercise, we used trigonometric identities and transformations to simplify the integration process.
A fundamental technique leveraged here is recognizing that \(\frac{d}{d\theta}( \sec \theta ) = \sec \theta \tan \theta\) and \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\). Knowing these derivatives can help in inversely identifying antiderivatives during integration.

• Trigonometric identities can simplify an integrand
• Understanding the derivatives of basic trigonometric functions aids in finding antiderivatives
• Substitutions, such as \(x = \tan \theta\), are often used to simplify integration

In more complex integrals, breaking down the integral using a combination of substitution and identities is a strategy to ease the solution, as seen in joining \(\sec \theta\) terms to \(\cos \theta\). This integral became much simpler after utilizing such integration techniques.

Definite Integrals

Definite integrals are used to find the exact area under a curve between two points. In this problem, definite integrals accompany trigonometric substitution, as we determine specific limits for \(\theta\). The limits of integration were initially \(x=0\) and \(x=\sqrt{3}\), corresponding to \(\theta=0\) and \(\theta=\frac{\pi}{3}\) after substitution.
After evaluating the transformed integral, it is crucial to compare it with the original integral to check the truth of the statement given in the exercise.

• Calculates area under a curve
• Involves limits of integration for exact measures
• Comparative analysis with original statements helps verify results

The evaluation of the integral from Step 1 resulted in \(\frac{\sqrt{3}}{2}\), whereas the original provided integral yielded a different value of \(-\frac{\sqrt{3}}{2}\). This discrepancy clearly demonstrated that the statement is false, highlighting the importance of careful analysis with definite integrals.