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

Lösung

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

\frac{2}{5} (\frac{t ^{2} + 1}{t})^{\frac{5}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende die Potenzregel an

= \frac{2}{5} u^{\frac{5}{2}}

Setze in

u = t + \frac{1}{t}

ein

= \frac{2}{5} (t + \frac{1}{t})^{\frac{5}{2}}

Füge

t + \frac{1}{t}

zusammen:

\frac{t ^{2} + 1}{t}

= \frac{2}{5} (\frac{t ^{2} + 1}{t})^{\frac{5}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (\frac{t ^{2} + 1}{t})^{\frac{5}{2}} + C

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