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

The integral from pi/4 to pi/2 of csc^4(x) is 4/3

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integral from pi/4 to pi/2 of csc^4(x)

\int_{\frac{π}{4}}^{\frac{π}{2}} csc^{4} (x) d x

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{2}} csc^{4} (x) d x

Apply Integral Reduction

= - [\frac{cos ( x ) csc ^{3} ( x )}{3}]_{\frac{π}{4}}^{\frac{π}{2}} + \frac{2}{3} \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} csc^{2} (x) d x

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

= - [\frac{cos ( x ) csc ^{3} ( x )}{3}]_{\frac{π}{4}}^{\frac{π}{2}} + \frac{2}{3} \cdot 1

Simplify

- [\frac{cos ( x ) csc ^{3} ( x )}{3}]_{\frac{π}{4}}^{\frac{π}{2}} + \frac{2}{3} \cdot 1 : - [\frac{1}{3} csc^{3} (x) cos (x)]_{\frac{π}{4}}^{\frac{π}{2}} + \frac{2}{3}

= - [\frac{1}{3} csc^{3} (x) cos (x)]_{\frac{π}{4}}^{\frac{π}{2}} + \frac{2}{3}

Compute the boundaries:

- \frac{2}{3}

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

Simplify

= \frac{4}{3}

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