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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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\int x (x + 1)^{7} d x

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\int \frac{1 ln ( x )}{x} d x

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