integral ((x-1))/((x^2+2x+3))

Lösung

integral

\frac{( x - 1 )}{( x ^{2} + 2 x + 3 )}

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x - 1}{x ^{2} + 2 x + 3} : \frac{x}{x ^{2} + 2 x + 3} - \frac{1}{x ^{2} + 2 x + 3}

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

\int \frac{x}{x ^{2} + 2 x + 3} d x = \frac{1}{2} ln x^{2} + 2 x + 3 - \frac{1}{2} arctan (\frac{1}{2} (x + 1))

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

= \frac{1}{2} ln x^{2} + 2 x + 3 - \frac{1}{2} arctan (\frac{1}{2} (x + 1)) - \frac{1}{2} arctan (\frac{x + 1}{2})

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{1}{x ( x ^{2} - 1 )}

in t e g r a l \frac{x ^{- 2} ( e ^{x} ) ^{3}}{3}

in t e g r a l \frac{1}{( 3 x ^{2} + 1 )}

in t e g r a l 4 x^{3} + 3 x^{2} + 2

in t e g r a l (1 - x^{2})^{\frac{3}{2}}