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

Lösung

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

\frac{π}{2 3} + ln (2)

Dezimale

1.60004 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

= [- 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}}

Vereinfache

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

Berechne die Grenzen:

- (- \frac{π}{2 3} - ln (2))

= - (- \frac{π}{2 3} - ln (2))

Vereinfache

= \frac{π}{2 3} + ln (2)

\int_{- 2}^{0} (4 x^{2} + 4 x) d x

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

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\int_{0}^{2 π} cos (x) + sin (x) d x