What is the integral from pi/8 to pi/2 of cot^3(x) ?

The integral from pi/8 to pi/2 of cot^3(x) is -ln(2)+(3+2sqrt(2)+ln(2-sqrt(2)))/2

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integral from pi/8 to pi/2 of cot^3(x)

\int_{\frac{π}{8}}^{\frac{π}{2}} cot^{3} (x) d x

- ln (2) + \frac{3 + 2 2 + ln ( 2 - 2 )}{2}

Decimal

1.95366 \dots

Solution steps

\int_{\frac{π}{8}}^{\frac{π}{2}} cot^{3} (x) d x

Apply Integral Reduction

= - [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{8}}^{\frac{π}{2}} - \int_{\frac{π}{8}}^{\frac{π}{2}} cot (x) d x

\int_{\frac{π}{8}}^{\frac{π}{2}} cot (x) d x = - \frac{1}{2} ln (2 - 2) + ln (2)

= - [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{8}}^{\frac{π}{2}} - (- \frac{1}{2} ln (2 - 2) + ln (2))

Simplify

- [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{8}}^{\frac{π}{2}} - (- \frac{1}{2} ln (2 - 2) + ln (2)) : - [\frac{1}{2} cot^{2} (x)]_{\frac{π}{8}}^{\frac{π}{2}} + \frac{1}{2} ln (2 - 2) - ln (2)

= - [\frac{1}{2} cot^{2} (x)]_{\frac{π}{8}}^{\frac{π}{2}} + \frac{1}{2} ln (2 - 2) - ln (2)

Compute the boundaries:

- \frac{3 + 2 2}{2}

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

Simplify

= - ln (2) + \frac{3 + 2 2 + ln ( 2 - 2 )}{2}

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What is the integral from pi/8 to pi/2 of cot^3(x) ?
The integral from pi/8 to pi/2 of cot^3(x) is -ln(2)+(3+2sqrt(2)+ln(2-sqrt(2)))/2