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 \(\int_{0}^{\sqrt{3}} \frac{d x}{\left(1+x^{2}\right)^{3 / 2}}=\int_{0}^{4 \pi / 3} \cos \theta d \theta .\)

Short 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-by-step solution

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 \(\int \frac{\sec^2(\theta)}{(\sec^2(\theta))^{3/2}} d \theta = \int \frac{d \theta}{\sec \theta} = \int \cos(\theta) d \theta \)

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.

Evaluate the transformed integral

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

Evaluate the provided integral

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

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