integral from 0 to 2pi of 7x^2sin(1/4 x)

Lösung

\int_{0}^{2 π} 7 x^{2} sin (\frac{1}{4} x) d x

448 (π - 2)

Dezimale

511.43350 \dots

Schritte zur Lösung

\int_{0}^{2 π} 7 x^{2} sin (\frac{1}{4} x) d x

Entferne die Konstante:

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

= 7 \cdot \int_{0}^{2 π} x^{2} sin (\frac{1}{4} x) d x

Wende U-Substitution an

= 7 \cdot \int_{0}^{\frac{π}{2}} 64 u^{2} sin (u) d u

Entferne die Konstante:

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

= 7 \cdot 64 \cdot \int_{0}^{\frac{π}{2}} u^{2} sin (u) d u

Wende die partielle Integration an

= 7 \cdot 64 [- u^{2} cos (u) - \int - 2 u cos (u) d u]_{0}^{\frac{π}{2}}

\int - 2 u cos (u) d u = - 2 (u sin (u) + cos (u))

= 7 \cdot 64 [- u^{2} cos (u) - (- 2 (u sin (u) + cos (u)))]_{0}^{\frac{π}{2}}

Vereinfache

= 448 [- u^{2} cos (u) + 2 (u sin (u) + cos (u))]_{0}^{\frac{π}{2}}

Berechne die Grenzen:

π - 2

= 448 (π - 2)

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