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Beliebt Rechnen >

integral of (x+3)/(x^2+8x-9)

Lösung

\int \frac{x + 3}{x ^{2} + 8 x - 9} d x

Lösung

ln ∣ x + 9 ∣ + 4 (\frac{1}{10} ln ∣ x - 1 ∣ - \frac{1}{10} ln ∣ x + 9 ∣) + C

Schritte zur Lösung

\int \frac{x + 3}{x ^{2} + 8 x - 9} d x

Faktorisiere

x^{2} + 8 x - 9 : (x - 1) (x + 9)

= \int \frac{x + 3}{( x - 1 ) ( x + 9 )} d x

Wende U-Substitution an

= \int \frac{u + 4}{u ( u + 10 )} d u

Multipliziere aus

\frac{u + 4}{u ( u + 10 )} : \frac{u}{u ( u + 10 )} + \frac{4}{u ( u + 10 )}

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{u}{u ( u + 10 )} d u + \int \frac{4}{u ( u + 10 )} d u

\int \frac{u}{u ( u + 10 )} d u = ln ∣ u + 10 ∣

\int \frac{4}{u ( u + 10 )} d u = 4 (\frac{1}{10} ln ∣ u ∣ - \frac{1}{10} ln ∣ u + 10 ∣)

= ln ∣ u + 10 ∣ + 4 (\frac{1}{10} ln ∣ u ∣ - \frac{1}{10} ln ∣ u + 10 ∣)

Setze in

u = x - 1

ein

= ln ∣ x - 1 + 10 ∣ + 4 (\frac{1}{10} ln ∣ x - 1 ∣ - \frac{1}{10} ln ∣ x - 1 + 10 ∣)

Vereinfache

ln ∣ x - 1 + 10 ∣ + 4 (\frac{1}{10} ln ∣ x - 1 ∣ - \frac{1}{10} ln ∣ x - 1 + 10 ∣) : ln ∣ x + 9 ∣ + 4 (\frac{1}{10} ln ∣ x - 1 ∣ - \frac{1}{10} ln ∣ x + 9 ∣)

= ln ∣ x + 9 ∣ + 4 (\frac{1}{10} ln ∣ x - 1 ∣ - \frac{1}{10} ln ∣ x + 9 ∣)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 9 ∣ + 4 (\frac{1}{10} ln ∣ x - 1 ∣ - \frac{1}{10} ln ∣ x + 9 ∣) + C

Graph

Beliebte Beispiele

(\partial)/(\partial y)(xysqrt(x^2+y^2))

\frac{\partial}{\partial y} (x y x^{2} + y^{2})

integral of xsqrt(3x^2+4)

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

integral of (5x)/(x^2+1)

\int \frac{5 x}{x ^{2} + 1} d x

sum from n=1 to infinity of 1/(n3^n)

n = 1 \sum \infty \frac{1}{n 3 ^{n}}

inverselaplace {1/(s^3)}

in v erse l a pl a ce {\frac{1}{s ^{3}}}