integral of (x-2)/(x^2+4x+13)

Lösung

\int \frac{x - 2}{x ^{2} + 4 x + 13} d x

\frac{1}{2} ln \frac{1}{9} (x + 2)^{2} + 1 - \frac{4}{3} arctan (\frac{1}{3} (x + 2)) + C

Schritte zur Lösung

\int \frac{x - 2}{x ^{2} + 4 x + 13} d x

Multipliziere aus

\frac{x - 2}{x ^{2} + 4 x + 13} : \frac{x}{x ^{2} + 4 x + 13} - \frac{2}{x ^{2} + 4 x + 13}

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} + 4 x + 13} d x - \int \frac{2}{x ^{2} + 4 x + 13} d x

\int \frac{x}{x ^{2} + 4 x + 13} d x = \frac{1}{2} ln \frac{( x + 2 ) ^{2}}{9} + 1 - \frac{2}{3} arctan (\frac{1}{3} (x + 2))

\int \frac{2}{x ^{2} + 4 x + 13} d x = \frac{2}{3} arctan (\frac{1}{3} (x + 2))

= \frac{1}{2} ln \frac{( x + 2 ) ^{2}}{9} + 1 - \frac{2}{3} arctan (\frac{1}{3} (x + 2)) - \frac{2}{3} arctan (\frac{1}{3} (x + 2))

Vereinfache

\frac{1}{2} ln \frac{( x + 2 ) ^{2}}{9} + 1 - \frac{2}{3} arctan (\frac{1}{3} (x + 2)) - \frac{2}{3} arctan (\frac{1}{3} (x + 2)) : \frac{1}{2} ln \frac{1}{9} (x + 2)^{2} + 1 - \frac{4}{3} arctan (\frac{1}{3} (x + 2))

= \frac{1}{2} ln \frac{1}{9} (x + 2)^{2} + 1 - \frac{4}{3} arctan (\frac{1}{3} (x + 2))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln \frac{1}{9} (x + 2)^{2} + 1 - \frac{4}{3} arctan (\frac{1}{3} (x + 2)) + C

\int 6 x^{2} - 4 x + 3 d x

\int 6 x^{6} + 3 x^{5} + 5 x - 4 d x

\int (x^{4} + 3) d x

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

\int \frac{e ^{x}}{- 8 + e ^{x}} d x