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Beliebt Rechnen >

integral of ((3x+2))/(sqrt(19-5x+x^2))

Lösung

\int \frac{( 3 x + 2 )}{19 - 5 x + x ^{2}} d x

Lösung

3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5) + C

Schritte zur Lösung

\int \frac{3 x + 2}{19 - 5 x + x ^{2}} d x

Multipliziere aus

\frac{3 x + 2}{19 - 5 x + x ^{2}} : \frac{3 x}{19 - 5 x + x ^{2}} + \frac{2}{19 - 5 x + x ^{2}}

= \int \frac{3 x}{19 - 5 x + x ^{2}} + \frac{2}{19 - 5 x + x ^{2}} 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{3 x}{19 - 5 x + x ^{2}} d x + \int \frac{2}{19 - 5 x + x ^{2}} d x

\int \frac{3 x}{19 - 5 x + x ^{2}} d x = 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{4 x ^{2} - 20 x + 76}{51})

\int \frac{2}{19 - 5 x + x ^{2}} d x = 2 ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)

= 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{4 x ^{2} - 20 x + 76}{51}) + 2 ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)

Vereinfache

3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{4 x ^{2} - 20 x + 76}{51}) + 2 ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76) : 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5)

= 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5)

Füge eine Konstante zur Lösung hinzu

= 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5) + C

Graph

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