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Beliebt Rechnen >

integral of (x+4)/(sqrt(6x-x^2))

Lösung

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

Lösung

- - x^{2} + 6 x + 7 arcsin (\frac{1}{3} x - 1) + C

Schritte zur Lösung

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

Vervollständige das Quadrat

6 x - x^{2} : - (x - 3)^{2} + 9

= \int \frac{x + 4}{- ( x - 3 ) ^{2} + 9} d x

Wende U-Substitution an

= \int \frac{u + 7}{- u ^{2} + 9} d u

Trigonometrische Substitution anwenden

= \int 3 sin (v) + 7 d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3 sin (v) d v + \int 7 d v

\int 3 sin (v) d v = - 3 cos (v)

\int 7 d v = 7 v

= - 3 cos (v) + 7 v

Ersetze zurück

= - 3 cos (arcsin (\frac{1}{3} (x - 3))) + 7 arcsin (\frac{1}{3} (x - 3))

Vereinfache

- 3 cos (arcsin (\frac{1}{3} (x - 3))) + 7 arcsin (\frac{1}{3} (x - 3)) : - - x^{2} + 6 x + 7 arcsin (\frac{1}{3} x - 1)

= - - x^{2} + 6 x + 7 arcsin (\frac{1}{3} x - 1)

Füge eine Konstante zur Lösung hinzu

= - - x^{2} + 6 x + 7 arcsin (\frac{1}{3} x - 1) + C

Graph

Beliebte Beispiele

integral of sqrt(x)sqrt(100-x)

\int x 100 - x d x

integral of sqrt(100-25x^2)

\int 100 - 25 x^{2} d x

integral of (x+5)/(sqrt(3x^2+30x-4))

\int \frac{x + 5}{3 x ^{2} + 30 x - 4} d x

integral of ((x^6+5x+3)/x)

\int (\frac{x ^{6} + 5 x + 3}{x}) d x

integral of 1/(x^3-16x)

\int \frac{1}{x ^{3} - 16 x} d x