de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of cos(x)cos(2nx)

Lösung

\int cos (x) cos (2 n x) d x

Lösung

\frac{1}{2} (\frac{1}{n} sin (2 n x) cos (x) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (2 n x + x) + \frac{2}{2 n - 1} sin (2 n x - x))) + C

Schritte zur Lösung

\int cos (x) cos (2 n x) d x

Wende integrale Substitution an

= \int \frac{cos ( n u ) cos ( \frac{u}{2} )}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cos (n u) cos (\frac{u}{2}) d u

Wende die partielle Integration an

= \frac{1}{2} (\frac{1}{n} sin (n u) cos (\frac{u}{2}) - \int - \frac{1}{2 n} sin (n u) sin (\frac{u}{2}) d u)

\int - \frac{1}{2 n} sin (n u) sin (\frac{u}{2}) d u = - \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n u + \frac{u}{2}) + \frac{2}{2 n - 1} sin (n u - \frac{u}{2}))

= \frac{1}{2} (\frac{1}{n} sin (n u) cos (\frac{u}{2}) - (- \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n u + \frac{u}{2}) + \frac{2}{2 n - 1} sin (n u - \frac{u}{2}))))

Setze in

u = 2 x

ein

= \frac{1}{2} (\frac{1}{n} sin (n \cdot 2 x) cos (\frac{2 x}{2}) - (- \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n \cdot 2 x + \frac{2 x}{2}) + \frac{2}{2 n - 1} sin (n \cdot 2 x - \frac{2 x}{2}))))

Vereinfache

\frac{1}{2} (\frac{1}{n} sin (n \cdot 2 x) cos (\frac{2 x}{2}) - (- \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n \cdot 2 x + \frac{2 x}{2}) + \frac{2}{2 n - 1} sin (n \cdot 2 x - \frac{2 x}{2})))) : \frac{1}{2} (\frac{1}{n} sin (2 n x) cos (x) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (2 n x + x) + \frac{2}{2 n - 1} sin (2 n x - x)))

= \frac{1}{2} (\frac{1}{n} sin (2 n x) cos (x) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (2 n x + x) + \frac{2}{2 n - 1} sin (2 n x - x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{1}{n} sin (2 n x) cos (x) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (2 n x + x) + \frac{2}{2 n - 1} sin (2 n x - x))) + C

Graph

Beliebte Beispiele

integral of cos(x)in(sin(x))

\int cos (x) in (sin (x)) d x

integral of 1/(1+sqrt(2x+3))

\int \frac{1}{1 + 2 x + 3} d x

integral of (x^2+5)2^{-x}

\int (x^{2} + 5) 2^{- x} d x

integral of 1/(x^2*sqrt(1+x^2))

\int \frac{1}{x ^{2} \cdot 1 + x ^{2}} d x

integral of x^3x^2

\int x^{3} x^{2} d x