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

Lösung

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

ln ∣ x + 3 ∣ + \frac{2}{x + 3} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x + 1}{( x + 3 ) ^{2}} : \frac{x}{( x + 3 ) ^{2}} + \frac{1}{( x + 3 ) ^{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{x}{( x + 3 ) ^{2}} d x + \int \frac{1}{( x + 3 ) ^{2}} d x

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

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

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

Vereinfache

ln ∣ x + 3 ∣ + \frac{3}{x + 3} - \frac{1}{x + 3} : ln ∣ x + 3 ∣ + \frac{2}{x + 3}

= ln ∣ x + 3 ∣ + \frac{2}{x + 3}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 3 ∣ + \frac{2}{x + 3} + C

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