integral of (x^2)/((x^2-1)^{3/2)}

Lösung

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

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

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int sec^{3} (u) cot^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{csc ^{2} ( u )}{cos ( u )} d u

Wende die partielle Integration an

= - csc (u) - \int - sec (u) d u

\int - sec (u) d u = - ln ∣ tan (u) + sec (u) ∣

= - csc (u) - (- ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arcsec (x)

ein

= - csc (arcsec (x)) - (- ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣)

Vereinfache

- csc (arcsec (x)) - (- ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣) : - \frac{x}{( x ^{2} - 1 ) ^{\frac{1}{2}}} + ln (x^{2} - 1)^{\frac{1}{2}} + x

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

Füge eine Konstante zur Lösung hinzu

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

t an g e n t f (x) = x^{2} - 5 x, at x = - 2

\int 9 x sin (x) cos (x) d x

t an g e n t f (x) = ∣ x ∣ 5 - x^{2}, (2, 2)

x \to 2 lim (3 x + 4 x)

\int_{2}^{4} \frac{2}{x x ^{2} - 4} d x