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Beliebt Rechnen >

integral of ((1-x))/(sqrt(x^2+4x+3))

Lösung

\int \frac{( 1 - x )}{x ^{2} + 4 x + 3} d x

Lösung

3 ln x + x^{2} + 4 x + 3 + 2 - x^{2} + 4 x + 3 + C

Schritte zur Lösung

\int \frac{1 - x}{x ^{2} + 4 x + 3} d x

Multipliziere aus

\frac{1 - x}{x ^{2} + 4 x + 3} : \frac{1}{x ^{2} + 4 x + 3} - \frac{x}{x ^{2} + 4 x + 3}

= \int \frac{1}{x ^{2} + 4 x + 3} - \frac{x}{x ^{2} + 4 x + 3} d 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{1}{x ^{2} + 4 x + 3} d x - \int \frac{x}{x ^{2} + 4 x + 3} d x

\int \frac{1}{x ^{2} + 4 x + 3} d x = ln x^{2} + 4 x + 3 + x + 2

\int \frac{x}{x ^{2} + 4 x + 3} d x = x^{2} + 4 x + 3 - 2 ln x^{2} + 4 x + 3 + x + 2

= ln x^{2} + 4 x + 3 + x + 2 - (x^{2} + 4 x + 3 - 2 ln x^{2} + 4 x + 3 + x + 2)

Vereinfache

= 3 ln x + x^{2} + 4 x + 3 + 2 - x^{2} + 4 x + 3

Füge eine Konstante zur Lösung hinzu

= 3 ln x + x^{2} + 4 x + 3 + 2 - x^{2} + 4 x + 3 + C

Graph

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