What is the integral from pi/4 to pi/2 of 8xcsc^2(x) ?

The integral from pi/4 to pi/2 of 8xcsc^2(x) is -8(-pi/4-1/2 ln(2))

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integral from pi/4 to pi/2 of 8xcsc^2(x)

\int_{\frac{π}{4}}^{\frac{π}{2}} 8 x csc^{2} (x) d x

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

Decimal

9.05577 \dots

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{2}} 8 x csc^{2} (x) d x

Take the constant out:

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

= 8 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} x csc^{2} (x) d x

Apply Integration By Parts

= 8 [- x cot (x) - \int - cot (x) d x]_{\frac{π}{4}}^{\frac{π}{2}}

\int - cot (x) d x = - ln ∣ sin (x) ∣

= 8 [- x cot (x) - (- ln ∣ sin (x) ∣)]_{\frac{π}{4}}^{\frac{π}{2}}

Simplify

= 8 [- x cot (x) + ln ∣ sin (x) ∣]_{\frac{π}{4}}^{\frac{π}{2}}

Compute the boundaries:

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

= 8 (- (- \frac{π}{4} - \frac{1}{2} ln (2)))

Simplify

= - 8 (- \frac{π}{4} - \frac{1}{2} ln (2))

What is the integral from pi/4 to pi/2 of 8xcsc^2(x) ?
The integral from pi/4 to pi/2 of 8xcsc^2(x) is -8(-pi/4-1/2 ln(2))