integral of (x+1)/(x^2+7x+10)

Lösung

\int \frac{x + 1}{x ^{2} + 7 x + 10} d x

2 (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{5}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x)) + C

Schritte zur Lösung

\int \frac{x + 1}{x ^{2} + 7 x + 10} d x

Vervollständige das Quadrat

x^{2} + 7 x + 10 : (x + \frac{7}{2})^{2} - \frac{9}{4}

= \int \frac{x + 1}{( x + \frac{7}{2} ) ^{2} - \frac{9}{4}} d x

Wende U-Substitution an

= \int \frac{2 ( 2 u - 5 )}{4 u ^{2} - 9} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{2 u - 5}{4 u ^{2} - 9} d u

Multipliziere aus

\frac{2 u - 5}{4 u ^{2} - 9} : \frac{2 u}{4 u ^{2} - 9} - \frac{5}{4 u ^{2} - 9}

Wende die Summenregel an:

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

= 2 (\int \frac{2 u}{4 u ^{2} - 9} d u - \int \frac{5}{4 u ^{2} - 9} d u)

\int \frac{2 u}{4 u ^{2} - 9} d u = \frac{1}{4} ln 4 u^{2} - 9

\int \frac{5}{4 u ^{2} - 9} d u = - \frac{5}{12} (ln \frac{2}{3} u + 1 - ln \frac{2}{3} u - 1)

= 2 (\frac{1}{4} ln 4 u^{2} - 9 - (- \frac{5}{12} (ln \frac{2}{3} u + 1 - ln \frac{2}{3} u - 1)))

Setze in

u = x + \frac{7}{2}

ein

= 2 (\frac{1}{4} ln 4 (x + \frac{7}{2})^{2} - 9 - (- \frac{5}{12} (ln \frac{2}{3} (x + \frac{7}{2}) + 1 - ln \frac{2}{3} (x + \frac{7}{2}) - 1)))

Vereinfache

2 (\frac{1}{4} ln 4 (x + \frac{7}{2})^{2} - 9 - (- \frac{5}{12} (ln \frac{2}{3} (x + \frac{7}{2}) + 1 - ln \frac{2}{3} (x + \frac{7}{2}) - 1))) : 2 (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{5}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x))

= 2 (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{5}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{5}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x)) + C

\int 12 - 8 x + x^{2} d x

\frac{d ^{2} S}{d x ^{2}} = S

x \to - \infty lim (- 3 x^{3} - x)

d er i v a t i v e 2 x^{2} + 7 x^{2} + 4 x

a re a 5 - x^{2}, 4 x^{2}