integral of ((5x^2-12))/((x^2-6x+13)^2)

Lösung

\int \frac{( 5 x ^{2} - 12 )}{( x ^{2} - 6 x + 13 ) ^{2}} d x

\frac{15}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{53}{16} arctan (\frac{1}{2} (x - 3)) - \frac{77}{32} sin (2 arctan (\frac{1}{2} (x - 3))) + C

Schritte zur Lösung

\int \frac{5 x ^{2} - 12}{( x ^{2} - 6 x + 13 ) ^{2}} d x

Multipliziere aus

\frac{5 x ^{2} - 12}{( x ^{2} - 6 x + 13 ) ^{2}} : \frac{5 x ^{2}}{( x ^{2} - 6 x + 13 ) ^{2}} - \frac{12}{( x ^{2} - 6 x + 13 ) ^{2}}

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{5 x ^{2}}{( x ^{2} - 6 x + 13 ) ^{2}} d x - \int \frac{12}{( x ^{2} - 6 x + 13 ) ^{2}} d x

\int \frac{5 x ^{2}}{( x ^{2} - 6 x + 13 ) ^{2}} d x = 5 (\frac{3}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{16} arctan (\frac{1}{2} (x - 3)) - \frac{13}{32} sin (2 arctan (\frac{1}{2} (x - 3))))

\int \frac{12}{( x ^{2} - 6 x + 13 ) ^{2}} d x = \frac{3}{4} (arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x - 3))))

= 5 (\frac{3}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{16} arctan (\frac{1}{2} (x - 3)) - \frac{13}{32} sin (2 arctan (\frac{1}{2} (x - 3)))) - \frac{3}{4} (arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x - 3))))

Vereinfache

5 (\frac{3}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{16} arctan (\frac{1}{2} (x - 3)) - \frac{13}{32} sin (2 arctan (\frac{1}{2} (x - 3)))) - \frac{3}{4} (arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x - 3)))) : \frac{15}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{53}{16} arctan (\frac{1}{2} (x - 3)) - \frac{77}{32} sin (2 arctan (\frac{1}{2} (x - 3)))

= \frac{15}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{53}{16} arctan (\frac{1}{2} (x - 3)) - \frac{77}{32} sin (2 arctan (\frac{1}{2} (x - 3)))

Füge eine Konstante zur Lösung hinzu

= \frac{15}{16} x sin (2 arctan (\frac{1}{2} (x - 3))) + \frac{53}{16} arctan (\frac{1}{2} (x - 3)) - \frac{77}{32} sin (2 arctan (\frac{1}{2} (x - 3))) + C

n = 1 \sum \infty \frac{1}{1 + n}

\frac{d}{d y} (y^{2} + 1)

x \to 3 lim (\frac{2}{3 - 2})

\frac{d}{d x} f (x) = x^{\frac{- 11}{12}}

\int \frac{tan ( ln ( 2 x + 5 ) )}{2 x + 5} d x