integral of x/(sqrt(9-(x-3)^2))

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u + 3}{9 - u ^{2}} d u

Trigonometrische Substitution anwenden

= \int 3 (sin (v) + 1) d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int sin (v) + 1 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

= 3 (\int sin (v) d v + \int 1 d v)

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

\int 1 d v = v

= 3 (- cos (v) + v)

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int x^{2} (x^{3} + 7)^{7} d x

\int 4 x + 16 d x

\int \frac{ln ( x )}{x 2 + ln ( x )} d x

\int x (2 x - 1)^{7} d x

\int \frac{5}{( 1 + x ^{2} )} d x