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Beliebt Rechnen >

integral of (2x+7)/(x^2+3x+1)

Lösung

\int \frac{2 x + 7}{x ^{2} + 3 x + 1} d x

Lösung

ln 4 x^{2} + 12 x + 4 - \frac{4}{5} (ln \frac{1}{5} (2 x + 3) + 1 - ln \frac{1}{5} (2 x + 3) - 1) + C

Schritte zur Lösung

\int \frac{2 x + 7}{x ^{2} + 3 x + 1} d x

Multipliziere aus

\frac{2 x + 7}{x ^{2} + 3 x + 1} : \frac{2 x}{x ^{2} + 3 x + 1} + \frac{7}{x ^{2} + 3 x + 1}

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{2 x}{x ^{2} + 3 x + 1} d x + \int \frac{7}{x ^{2} + 3 x + 1} d x

\int \frac{2 x}{x ^{2} + 3 x + 1} d x = ln 4 x^{2} + 12 x + 4 + \frac{3}{5} (ln \frac{2 x + 3}{5} + 1 - ln \frac{2 x + 3}{5} - 1)

\int \frac{7}{x ^{2} + 3 x + 1} d x = - \frac{7}{5} (ln \frac{2 x + 3}{5} + 1 - ln \frac{2 x + 3}{5} - 1)

= ln 4 x^{2} + 12 x + 4 + \frac{3}{5} (ln \frac{2 x + 3}{5} + 1 - ln \frac{2 x + 3}{5} - 1) - \frac{7}{5} (ln \frac{2 x + 3}{5} + 1 - ln \frac{2 x + 3}{5} - 1)

Vereinfache

ln 4 x^{2} + 12 x + 4 + \frac{3}{5} (ln \frac{2 x + 3}{5} + 1 - ln \frac{2 x + 3}{5} - 1) - \frac{7}{5} (ln \frac{2 x + 3}{5} + 1 - ln \frac{2 x + 3}{5} - 1) : ln 4 x^{2} + 12 x + 4 - \frac{4}{5} (ln \frac{1}{5} (2 x + 3) + 1 - ln \frac{1}{5} (2 x + 3) - 1)

= ln 4 x^{2} + 12 x + 4 - \frac{4}{5} (ln \frac{1}{5} (2 x + 3) + 1 - ln \frac{1}{5} (2 x + 3) - 1)

Füge eine Konstante zur Lösung hinzu

= ln 4 x^{2} + 12 x + 4 - \frac{4}{5} (ln \frac{1}{5} (2 x + 3) + 1 - ln \frac{1}{5} (2 x + 3) - 1) + C

Graph

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