What is the integral from pi/4 to pi/2 of 9cot^3(x) ?

The integral from pi/4 to pi/2 of 9cot^3(x) is 9(1/2-1/2 ln(2))

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integral from pi/4 to pi/2 of 9cot^3(x)

\int_{\frac{π}{4}}^{\frac{π}{2}} 9 cot^{3} (x) d x

9 (\frac{1}{2} - \frac{1}{2} ln (2))

Decimal Notation

1.38083 \dots

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{2}} 9 cot^{3} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 9 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} cot^{3} (x) d x

Apply Integral Reduction

= 9 (- [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{4}}^{\frac{π}{2}} - \int_{\frac{π}{4}}^{\frac{π}{2}} cot (x) d x)

\int_{\frac{π}{4}}^{\frac{π}{2}} cot (x) d x = \frac{1}{2} ln (2)

= 9 (- [\frac{cot ^{2} ( x )}{2}]_{\frac{π}{4}}^{\frac{π}{2}} - \frac{1}{2} ln (2))

Compute the boundaries:

- \frac{1}{2}

= 9 (- (- \frac{1}{2}) - \frac{1}{2} ln (2))

Simplify

= 9 (\frac{1}{2} - \frac{1}{2} ln (2))

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What is the integral from pi/4 to pi/2 of 9cot^3(x) ?
The integral from pi/4 to pi/2 of 9cot^3(x) is 9(1/2-1/2 ln(2))