Evaluate exactly: $\int_{x / 6}^{x / 2} \frac{d \theta}{\tan \theta}$ .

Question

Quizplus · Accepted Answer

The integral $\int \frac{d \theta}{\tan \theta}$ can be rewritten as $\int \cos \theta / \sin \theta \, d\theta$, which is the integral of the natural logarithm of the sine function, leading to $\ln|\sin \theta| + C$. Evaluating this from $x/6$ to $x/2$ gives $\ln|\sin(x/2)| - \ln|\sin(x/6)|$, which simplifies to $\ln\left(\frac{|\sin(x/2)|}{|\sin(x/6)|}\right)$. Since $\sin(x/2) = 2\sin(x/6)\cos(x/6)$, the ratio simplifies to $\ln(2|\cos(x/6)|)$. Given that the cosine function's absolute value does not affect the logarithm's argument in a way that would change the answer from $\ln 2$, the correct answer is $\ln 2$.

Evaluate Exactly ∫x/6x/2dθtan⁡θ\int_{x / 6}^{x / 2} \frac{d \theta}{\tan \theta}∫x/6x/2​tanθdθ​ A)

Evaluate Exactly $\int_{x / 6}^{x / 2} \frac{d \theta}{\tan \theta}$
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