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

Lösung

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

- \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x + 7}{x ^{2} + 2 x} : \frac{x}{x ^{2} + 2 x} + \frac{7}{x ^{2} + 2 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{x}{x ^{2} + 2 x} d x + \int \frac{7}{x ^{2} + 2 x} d x

\int \frac{x}{x ^{2} + 2 x} d x = ln ∣ x + 2 ∣

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

= ln ∣ x + 2 ∣ - 7 (\frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣)

Vereinfache

ln ∣ x + 2 ∣ - 7 (\frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣) : - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣

= - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣ + C

\int (3 x^{3} + 4 x - 7) d x

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\int \frac{e ^{x}}{x ^{2}} \cdot e^{x} + 2 x d x