integral from 0 to 2 of |5xcos(pix)|

Lösung

\int_{0}^{2} ∣ 5 x cos (π x) ∣ d x

\frac{20}{π}

Dezimale

6.36619 \dots

Schritte zur Lösung

\int_{0}^{2} ∣ 5 x cos (π x) ∣ d x

Vereinfache

∣ 5 x cos (π x) ∣ : 5 ∣ x ∣ ∣ cos (π x) ∣

= \int_{0}^{2} 5 ∣ x ∣ ∣ cos (π x) ∣ d x

Entferne die Extremwerte

= \int_{0}^{\frac{1}{2}} 5 x cos (π x) d x + \int_{\frac{1}{2}}^{\frac{3}{2}} 5 x (- cos (π x)) d x + \int_{\frac{3}{2}}^{2} 5 x cos (π x) d x

\int_{0}^{\frac{1}{2}} 5 x cos (π x) d x = 5 (\frac{1}{2 π} - \frac{1}{π ^{2}})

\int_{\frac{1}{2}}^{\frac{3}{2}} 5 x (- cos (π x)) d x = \frac{10}{π}

\int_{\frac{3}{2}}^{2} 5 x cos (π x) d x = 5 (\frac{1}{π ^{2}} + \frac{3}{2 π})

= 5 (\frac{1}{2 π} - \frac{1}{π ^{2}}) + \frac{10}{π} + 5 (\frac{1}{π ^{2}} + \frac{3}{2 π})

Vereinfache

5 (\frac{1}{2 π} - \frac{1}{π ^{2}}) + \frac{10}{π} + 5 (\frac{1}{π ^{2}} + \frac{3}{2 π}) : \frac{20}{π}

= \frac{20}{π}

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