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

Lösung

integral

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

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

Schritte zur Lösung

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

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

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

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

\int 1 d x = x

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{- 1}{y}

f (x) = \frac{1}{( 1 - x ) ^{\frac{1}{2}}}

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

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

in t e g r a l \frac{2 x}{x + 2}