What is the integral from pi/6 to pi/3 of cot^3(x) ?

The integral from pi/6 to pi/3 of cot^3(x) is 4/3-1/2 ln(3)

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integral from pi/6 to pi/3 of cot^3(x)

\int_{\frac{π}{6}}^{\frac{π}{3}} cot^{3} (x) d x

\frac{4}{3} - \frac{1}{2} ln (3)

Decimal

0.78402 \dots

Solution steps

\int_{\frac{π}{6}}^{\frac{π}{3}} cot^{3} (x) d x

Apply Integral Reduction

= - [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{6}}^{\frac{π}{3}} - \int_{\frac{π}{6}}^{\frac{π}{3}} cot (x) d x

\int_{\frac{π}{6}}^{\frac{π}{3}} cot (x) d x = \frac{1}{2} ln (3)

= - [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{6}}^{\frac{π}{3}} - \frac{1}{2} ln (3)

Compute the boundaries:

- \frac{4}{3}

= - (- \frac{4}{3}) - \frac{1}{2} ln (3)

Simplify

= \frac{4}{3} - \frac{1}{2} ln (3)

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What is the integral from pi/6 to pi/3 of cot^3(x) ?
The integral from pi/6 to pi/3 of cot^3(x) is 4/3-1/2 ln(3)