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

The integral from pi/4 to pi/2 of xcsc^2(x) is pi/4+1/2 ln(2)

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

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

\frac{π}{4} + \frac{1}{2} ln (2)

Solution steps

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

Apply Integration By Parts

: [- x cot (x) - \int - cot (x) d x]_{\frac{π}{4}}^{\frac{π}{2}}

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

\int - cot (x) d x = - ln ∣ sin (x) ∣

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

Simplify

= [- x cot (x) + ln ∣ sin (x) ∣]_{\frac{π}{4}}^{\frac{π}{2}}

Compute the boundaries:

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

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

Simplify

= \frac{π}{4} + \frac{1}{2} ln (2)

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What is the integral from pi/4 to pi/2 of xcsc^2(x) ?
The integral from pi/4 to pi/2 of xcsc^2(x) is pi/4+1/2 ln(2)