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

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

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

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

- 1

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{2}^{1} - \frac{1}{2} d u

= - 2 \cdot \int_{2}^{1} - \frac{1}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - 2 (- \int_{1}^{2} - \frac{1}{2} d u)

Integral of a constant:

\int a d x = a x

= - 2 (- [(- \frac{1}{2}) u]_{1}^{2})

Simplify

= 2 [- \frac{1}{2} u]_{1}^{2}

Compute the boundaries:

- \frac{1}{2}

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

Simplify

= - 1

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