integral of 1/(cos^2(x)+5cos(x)+6)

Lösung

\int \frac{1}{cos ^{2} ( x ) + 5 cos ( x ) + 6} d x

\frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) - \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} )}{2}) + C

Schritte zur Lösung

\int \frac{1}{cos ^{2} ( x ) + 5 cos ( x ) + 6} d x

Ermittle den Partialbruch von

\frac{1}{cos ^{2} ( x ) + 5 cos ( x ) + 6} : \frac{1}{cos ( x ) + 2} - \frac{1}{cos ( x ) + 3}

= \int \frac{1}{cos ( x ) + 2} - \frac{1}{cos ( x ) + 3} d x

Wende die Summenregel an:

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

= \int \frac{1}{cos ( x ) + 2} d x - \int \frac{1}{cos ( x ) + 3} d x

\int \frac{1}{cos ( x ) + 2} d x = \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

\int \frac{1}{cos ( x ) + 3} d x = \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} )}{2})

= \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) - \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} )}{2})

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) - \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} )}{2}) + C

\int 10 x^{3} + 2 x^{2} + \frac{1}{2} x + 3 d x

\frac{d y}{d x} = (y - 5)^{2}

\frac{d y}{d x}

d er i v a t i v e \frac{- 14 x}{12 y}

a re a y = \frac{1}{2} x^{2}, y = - x^{2} + 6