integral of sqrt(x^2-6x+10)

Lösung

\int x^{2} - 6 x + 10 d x

\frac{1}{2} (x - 3) x^{2} - 6 x + 10 + \frac{1}{2} ln x - 3 + x^{2} - 6 x + 10 + C

Schritte zur Lösung

\int x^{2} - 6 x + 10 d x

Vervollständige das Quadrat

x^{2} - 6 x + 10 : (x - 3)^{2} + 1

= \int (x - 3)^{2} + 1 d x

Wende U-Substitution an

= \int u^{2} + 1 d u

Trigonometrische Substitution anwenden

= \int sec^{3} (v) d v

Wende integrale Reduktion an

= \frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

= \frac{sec ^{2} ( arctan ( x - 3 ) ) sin ( arctan ( x - 3 ) )}{2} + \frac{1}{2} ln ∣ tan (arctan (x - 3)) + sec (arctan (x - 3)) ∣

Vereinfache

\frac{sec ^{2} ( arctan ( x - 3 ) ) sin ( arctan ( x - 3 ) )}{2} + \frac{1}{2} ln ∣ tan (arctan (x - 3)) + sec (arctan (x - 3)) ∣ : \frac{1}{2} (x - 3) x^{2} - 6 x + 10 + \frac{1}{2} ln x - 3 + x^{2} - 6 x + 10

= \frac{1}{2} (x - 3) x^{2} - 6 x + 10 + \frac{1}{2} ln x - 3 + x^{2} - 6 x + 10

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x - 3) x^{2} - 6 x + 10 + \frac{1}{2} ln x - 3 + x^{2} - 6 x + 10 + C

y^{^{''}} - 5 y^{^{'}} = (8 t - 7) sin (2 t)

x \to 7 lim (x (x - 2))

\frac{d}{d x} (cos (5 x^{2} + 7 x - 1))

d er i v a t i v e 2 x \cdot cos (x^{2})

\int \frac{1}{1 0 ^{x}} d x