integral of ((8x-3)/(sqrt(12x-4x^2-5)))

Lösung

\int (\frac{8 x - 3}{12 x - 4 x ^{2} - 5}) d x

\frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2})) + C

Schritte zur Lösung

\int \frac{8 x - 3}{12 x - 4 x ^{2} - 5} d x

Wende U-Substitution an

= \int \frac{u}{2 - u ^{2} + 18 u - 17} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u}{- u ^{2} + 18 u - 17} d u

Vervollständige das Quadrat

- u^{2} + 18 u - 17 : - (u - 9)^{2} + 64

= \frac{1}{2} \cdot \int \frac{u}{- ( u - 9 ) ^{2} + 64} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int \frac{v + 9}{- v ^{2} + 64} d v

Trigonometrische Substitution anwenden

= \frac{1}{2} \cdot \int 8 sin (w) + 9 d w

Wende die Summenregel an:

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

= \frac{1}{2} (\int 8 sin (w) d w + \int 9 d w)

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

\int 9 d w = 9 w

= \frac{1}{2} (- 8 cos (w) + 9 w)

Ersetze zurück

= \frac{1}{2} (- 8 cos (arcsin (\frac{1}{8} (8 x - 3 - 9))) + 9 arcsin (\frac{1}{8} (8 x - 3 - 9)))

Vereinfache

\frac{1}{2} (- 8 cos (arcsin (\frac{1}{8} (8 x - 3 - 9))) + 9 arcsin (\frac{1}{8} (8 x - 3 - 9))) : \frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2}))

= \frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2})) + C

\int \frac{sin ( x )}{1 + 2 cos ( x )} d x

l a pl a ce t r an s f or m cos^{2} (3 t)

\int e^{- t (s + 1)} d t

in v erse l a pl a ce \frac{e ^{- s}}{1 - s}

t a y l or sin (x), (\frac{π}{4})