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Beliebt Rechnen >

integral of (x+9)/(sqrt(64-(x-8)^2))

Lösung

\int \frac{x + 9}{64 - ( x - 8 ) ^{2}} d x

Lösung

- - x^{2} + 16 x + 17 arcsin (\frac{1}{8} x - 1) + C

Schritte zur Lösung

\int \frac{x + 9}{64 - ( x - 8 ) ^{2}} d x

Wende U-Substitution an

= \int \frac{u + 17}{64 - u ^{2}} d u

Trigonometrische Substitution anwenden

= \int 8 sin (v) + 17 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 8 sin (v) d v + \int 17 d v

\int 8 sin (v) d v = - 8 cos (v)

\int 17 d v = 17 v

= - 8 cos (v) + 17 v

Ersetze zurück

= - 8 cos (arcsin (\frac{1}{8} (x - 8))) + 17 arcsin (\frac{1}{8} (x - 8))

Vereinfache

- 8 cos (arcsin (\frac{1}{8} (x - 8))) + 17 arcsin (\frac{1}{8} (x - 8)) : - - x^{2} + 16 x + 17 arcsin (\frac{1}{8} x - 1)

= - - x^{2} + 16 x + 17 arcsin (\frac{1}{8} x - 1)

Füge eine Konstante zur Lösung hinzu

= - - x^{2} + 16 x + 17 arcsin (\frac{1}{8} x - 1) + C

Graph

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