What is the integral from pi/6 to pi/2 of xcsc^2(x) ?

The integral from pi/6 to pi/2 of xcsc^2(x) is pi/(2sqrt(3))+ln(2)

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integral from pi/6 to pi/2 of xcsc^2(x)

\int_{\frac{π}{6}}^{\frac{π}{2}} x csc^{2} (x) d x

\frac{π}{2 3} + ln (2)

Decimal Notation

1.60004 \dots

Solution steps

\int_{\frac{π}{6}}^{\frac{π}{2}} x csc^{2} (x) d x

Apply Integration By Parts

: [- x cot (x) - \int - cot (x) d x]_{\frac{π}{6}}^{\frac{π}{2}}

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

\int - cot (x) d x = - ln ∣ sin (x) ∣

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

Simplify

= [- x cot (x) + ln ∣ sin (x) ∣]_{\frac{π}{6}}^{\frac{π}{2}}

Compute the boundaries:

- (- \frac{π}{2 3} - ln (2))

= - (- \frac{π}{2 3} - ln (2))

Simplify

= \frac{π}{2 3} + ln (2)

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\int_{0}^{1} 16 x^{3} ln (x) d x

\int_{\frac{π}{3}}^{x} tan^{2} (t) d t

\int_{0}^{2 π} cos (x) + sin (x) d x

What is the integral from pi/6 to pi/2 of xcsc^2(x) ?
The integral from pi/6 to pi/2 of xcsc^2(x) is pi/(2sqrt(3))+ln(2)