de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of sin^7(x)cos^7(x)

Lösung

\int sin^{7} (x) cos^{7} (x) d x

Lösung

- \frac{cos ^{8} ( x )}{8} + \frac{3 cos ^{10} ( x )}{10} - \frac{cos ^{12} ( x )}{4} + \frac{cos ^{14} ( x )}{14} + C

Schritte zur Lösung

\int sin^{7} (x) cos^{7} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - cos^{2} (x))^{3} sin (x) cos^{7} (x) d x

Wende U-Substitution an

= \int - u^{7} (1 - u^{2})^{3} d u

Multipliziere aus

- u^{7} (1 - u^{2})^{3} : - u^{7} + 3 u^{9} - 3 u^{11} + u^{13}

= \int - u^{7} + 3 u^{9} - 3 u^{11} + u^{13} 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 u^{7} d u + \int 3 u^{9} d u - \int 3 u^{11} d u + \int u^{13} d u

\int u^{7} d u = \frac{u ^{8}}{8}

\int 3 u^{9} d u = \frac{3 u ^{10}}{10}

\int 3 u^{11} d u = \frac{u ^{12}}{4}

\int u^{13} d u = \frac{u ^{14}}{14}

= - \frac{u ^{8}}{8} + \frac{3 u ^{10}}{10} - \frac{u ^{12}}{4} + \frac{u ^{14}}{14}

Setze in

u = cos (x)

ein

= - \frac{cos ^{8} ( x )}{8} + \frac{3 cos ^{10} ( x )}{10} - \frac{cos ^{12} ( x )}{4} + \frac{cos ^{14} ( x )}{14}

Füge eine Konstante zur Lösung hinzu

= - \frac{cos ^{8} ( x )}{8} + \frac{3 cos ^{10} ( x )}{10} - \frac{cos ^{12} ( x )}{4} + \frac{cos ^{14} ( x )}{14} + C

Graph

Beliebte Beispiele

integral of 1/(sin(x)+cos(x)+1)

\int \frac{1}{sin ( x ) + cos ( x ) + 1} d x

integral of 6/(x^2+6x)

\int \frac{6}{x ^{2} + 6 x} d x

integral of 1/(x(ln(x^3)))

\int \frac{1}{x ( ln ( x ^{3} ) )} d x

sum from n=1 to infinity of 5/(n^2+64)

n = 1 \sum \infty \frac{5}{n ^{2} + 64}

integral from 1 to 6x of 7/t

\int_{1}^{6 x} \frac{7}{t} d t