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Beliebt Rechnen >

integral of (3x^2+4x-6)/(x^2+16x+64)

Lösung

\int \frac{3 x ^{2} + 4 x - 6}{x ^{2} + 16 x + 64} d x

Lösung

3 x - 44 ln ∣ x + 8 ∣ - \frac{154}{x + 8} + 24 + C

Schritte zur Lösung

\int \frac{3 x ^{2} + 4 x - 6}{x ^{2} + 16 x + 64} d x

Multipliziere aus

\frac{3 x ^{2} + 4 x - 6}{x ^{2} + 16 x + 64} : \frac{3 x ^{2}}{x ^{2} + 16 x + 64} + \frac{4 x}{x ^{2} + 16 x + 64} - \frac{6}{x ^{2} + 16 x + 64}

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{3 x ^{2}}{x ^{2} + 16 x + 64} d x + \int \frac{4 x}{x ^{2} + 16 x + 64} d x - \int \frac{6}{x ^{2} + 16 x + 64} d x

\int \frac{3 x ^{2}}{x ^{2} + 16 x + 64} d x = 3 (x + 8 - 16 ln ∣ x + 8 ∣ - \frac{64}{x + 8})

\int \frac{4 x}{x ^{2} + 16 x + 64} d x = 4 (ln ∣ x + 8 ∣ + \frac{8}{x + 8})

\int \frac{6}{x ^{2} + 16 x + 64} d x = - \frac{6}{x + 8}

= 3 (x + 8 - 16 ln ∣ x + 8 ∣ - \frac{64}{x + 8}) + 4 (ln ∣ x + 8 ∣ + \frac{8}{x + 8}) - (- \frac{6}{x + 8})

Vereinfache

3 (x + 8 - 16 ln ∣ x + 8 ∣ - \frac{64}{x + 8}) + 4 (ln ∣ x + 8 ∣ + \frac{8}{x + 8}) - (- \frac{6}{x + 8}) : 3 x - 44 ln ∣ x + 8 ∣ - \frac{154}{x + 8} + 24

= 3 x - 44 ln ∣ x + 8 ∣ - \frac{154}{x + 8} + 24

Füge eine Konstante zur Lösung hinzu

= 3 x - 44 ln ∣ x + 8 ∣ - \frac{154}{x + 8} + 24 + C

Graph

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