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Beliebt Rechnen >

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

Lösung

\int \frac{t ^{3}}{t ^{2} - 9} d t

Lösung

\frac{1}{3} (t^{2} - 9)^{\frac{3}{2}} + 9 t^{2} - 9 + C

Schritte zur Lösung

\int \frac{t ^{3}}{t ^{2} - 9} d t

Wende U-Substitution an

= \int u^{2} + 9 d u

Wende die Summenregel an:

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

= \int u^{2} d u + \int 9 d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 9 d u = 9 u

= \frac{u ^{3}}{3} + 9 u

Setze in

u = t^{2} - 9

ein

= \frac{( t ^{2} - 9 ) ^{3}}{3} + 9 t^{2} - 9

Vereinfache

\frac{( t ^{2} - 9 ) ^{3}}{3} + 9 t^{2} - 9 : \frac{1}{3} (t^{2} - 9)^{\frac{3}{2}} + 9 t^{2} - 9

= \frac{1}{3} (t^{2} - 9)^{\frac{3}{2}} + 9 t^{2} - 9

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (t^{2} - 9)^{\frac{3}{2}} + 9 t^{2} - 9 + C

Graph

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