integral from 0 to 2 of |5tcos(pit)|

Lösung

\int_{0}^{2} ∣ 5 t cos (π t) ∣ d t

\frac{20}{π}

Dezimale

6.36619 \dots

Schritte zur Lösung

\int_{0}^{2} ∣ 5 t cos (π t) ∣ d t

Vereinfache

∣ 5 t cos (π t) ∣ : 5 ∣ t ∣ ∣ cos (π t) ∣

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

Entferne die Extremwerte

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

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

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

\int_{\frac{3}{2}}^{2} 5 t cos (π t) d t = 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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