integral from 2 to 3 of cos^5(y)

Lösung

\int_{2}^{3} cos^{5} (y) d y

- sin (2) + sin (3) + \frac{2 ( sin ^{3} ( 2 ) - sin ^{3} ( 3 ) )}{3} - \frac{sin ^{5} ( 2 ) - sin ^{5} ( 3 )}{5}

Dezimale

- 0.39314 \dots

Schritte zur Lösung

\int_{2}^{3} cos^{5} (y) d y

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{2}^{3} (1 - sin^{2} (y))^{2} cos (y) d y

Wende U-Substitution an

= \int_{s i n (2)}^{s i n (3)} (1 - u^{2})^{2} d u

Multipliziere aus

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

= \int_{s i n (2)}^{s i n (3)} 1 - 2 u^{2} + u^{4} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{s i n (3)}^{s i n (2)} 1 - 2 u^{2} + u^{4} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int_{s i n (3)}^{s i n (2)} 1 d u - \int_{s i n (3)}^{s i n (2)} 2 u^{2} d u + \int_{s i n (3)}^{s i n (2)} u^{4} d u)

\int_{s i n (3)}^{s i n (2)} 1 d u = sin (2) - sin (3)

\int_{s i n (3)}^{s i n (2)} 2 u^{2} d u = \frac{2 ( sin ^{3} ( 2 ) - sin ^{3} ( 3 ) )}{3}

\int_{s i n (3)}^{s i n (2)} u^{4} d u = \frac{sin ^{5} ( 2 ) - sin ^{5} ( 3 )}{5}

= - (sin (2) - sin (3) - \frac{2 ( sin ^{3} ( 2 ) - sin ^{3} ( 3 ) )}{3} + \frac{sin ^{5} ( 2 ) - sin ^{5} ( 3 )}{5})

Vereinfache

= - sin (2) + sin (3) + \frac{2 ( sin ^{3} ( 2 ) - sin ^{3} ( 3 ) )}{3} - \frac{sin ^{5} ( 2 ) - sin ^{5} ( 3 )}{5}

\int_{0}^{1000} e^{4 x} - 5 d x

\int_{0}^{e} \frac{1}{2 x + e} d x

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