Get the solution to "Integration techniques Use the methods introduced evaluate the..."

Isa Trevino

Answered question

2021-11-05

Integration techniques Use the methods introduced evaluate the following integrals.

\int_{π}^{2 π} \cot \frac{x}{3} d x

Layton

Skilled2021-11-06Added 89 answers

Step 1
The given integral is

\int_{π}^{2 π} \cot \frac{x}{3} d x

.
Evaluate the above integral as follows.
Step 2

\int_{π}^{2 π} \cot \frac{x}{3} d x = \int_{π}^{2 π} \frac{\cos \frac{x}{3}}{\sin \frac{x}{3}} d x

= 3 \int_{π}^{2 π} \frac{\frac{1}{3} \cos \frac{x}{3}}{\sin \frac{x}{3}} d x

= 3 \int_{π}^{2 π} \frac{d (\sin \frac{x}{3})}{\sin \frac{x}{3}}

= 3 {[\ln (\sin \frac{x}{3})]}_{π}^{2 π}

= 3 [\ln (\sin \frac{2 π}{3}) - \ln (\sin \frac{π}{3})]

= 3 [\ln (\frac{\sqrt{3}}{2}) - \ln (\frac{\sqrt{3}}{2})]

=3(0)
=0
Thus,

\int_{π}^{2 π} \cot \frac{x}{3} d x = 0

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