integral from 2 to 4 of sin^7(y)

Lösung

\int_{2}^{4} sin^{7} (y) d y

cos (2) - cos (4) - cos^{3} (2) + cos^{3} (4) + \frac{3 ( cos ^{5} ( 2 ) - cos ^{5} ( 4 ) )}{5} - \frac{cos ^{7} ( 2 ) - cos ^{7} ( 4 )}{7}

Dezimale

0.08742 \dots

Schritte zur Lösung

\int_{2}^{4} sin^{7} (y) d y

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{2}^{4} (1 - cos^{2} (y))^{3} sin (y) d y

Wende U-Substitution an

= \int_{c o s (2)}^{c o s (4)} - (1 - u^{2})^{3} d u

Multipliziere aus

- (1 - u^{2})^{3} : - 1 + 3 u^{2} - 3 u^{4} + u^{6}

= \int_{c o s (2)}^{c o s (4)} - 1 + 3 u^{2} - 3 u^{4} + u^{6} d u

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

= - \int_{c o s (4)}^{c o s (2)} - 1 + 3 u^{2} - 3 u^{4} + u^{6} 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_{c o s (4)}^{c o s (2)} 1 d u + \int_{c o s (4)}^{c o s (2)} 3 u^{2} d u - \int_{c o s (4)}^{c o s (2)} 3 u^{4} d u + \int_{c o s (4)}^{c o s (2)} u^{6} d u)

\int_{c o s (4)}^{c o s (2)} 1 d u = cos (2) - cos (4)

\int_{c o s (4)}^{c o s (2)} 3 u^{2} d u = cos^{3} (2) - cos^{3} (4)

\int_{c o s (4)}^{c o s (2)} 3 u^{4} d u = \frac{3 ( cos ^{5} ( 2 ) - cos ^{5} ( 4 ) )}{5}

\int_{c o s (4)}^{c o s (2)} u^{6} d u = \frac{cos ^{7} ( 2 ) - cos ^{7} ( 4 )}{7}

= - (- (cos (2) - cos (4)) + cos^{3} (2) - cos^{3} (4) - \frac{3 ( cos ^{5} ( 2 ) - cos ^{5} ( 4 ) )}{5} + \frac{cos ^{7} ( 2 ) - cos ^{7} ( 4 )}{7})

Vereinfache

= cos (2) - cos (4) - cos^{3} (2) + cos^{3} (4) + \frac{3 ( cos ^{5} ( 2 ) - cos ^{5} ( 4 ) )}{5} - \frac{cos ^{7} ( 2 ) - cos ^{7} ( 4 )}{7}

\int_{1}^{2} e^{x} ln (x) d x

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