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

The integral from pi/3 to pi/2 of csc^2(x/2) is 2(-1+sqrt(3))

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

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

2 (- 1 + 3)

Solution steps

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

Apply u-substitution

: \int_{\frac{π}{6}}^{\frac{π}{4}} csc^{2} (u) \cdot 2 d u

= \int_{\frac{π}{6}}^{\frac{π}{4}} csc^{2} (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int_{\frac{π}{6}}^{\frac{π}{4}} csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= 2 [- cot (u)]_{\frac{π}{6}}^{\frac{π}{4}}

Compute the boundaries:

- 1 + 3

= 2 (- 1 + 3)

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What is the integral from pi/3 to pi/2 of csc^2(x/2) ?
The integral from pi/3 to pi/2 of csc^2(x/2) is 2(-1+sqrt(3))