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Beliebt Rechnen >

integral of ((t^m-t^n)^2)/(sqrt(t))

Lösung

\int \frac{( t ^{m} - t ^{n} ) ^{2}}{t} d t

Lösung

\frac{2 t ^{\frac{4 m + 1}{2}}}{4 m + 1} - \frac{4 t ^{\frac{2 m + 2 n + 1}{2}}}{2 m + 2 n + 1} + \frac{2 t ^{\frac{4 n + 1}{2}}}{4 n + 1} + C

Schritte zur Lösung

\int \frac{( t ^{m} - t ^{n} ) ^{2}}{t} d t

Multipliziere aus

\frac{( t ^{m} - t ^{n} ) ^{2}}{t} : t^{\frac{4 m - 1}{2}} - 2 t^{\frac{2 m + 2 n - 1}{2}} + t^{\frac{4 n - 1}{2}}

= \int t^{\frac{4 m - 1}{2}} - 2 t^{\frac{2 m + 2 n - 1}{2}} + t^{\frac{4 n - 1}{2}} d t

Wende die Summenregel an:

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

= \int t^{\frac{4 m - 1}{2}} d t - \int 2 t^{\frac{2 m + 2 n - 1}{2}} d t + \int t^{\frac{4 n - 1}{2}} d t

\int t^{\frac{4 m - 1}{2}} d t = \frac{2 t ^{\frac{4 m + 1}{2}}}{4 m + 1}

\int 2 t^{\frac{2 m + 2 n - 1}{2}} d t = \frac{4 t ^{\frac{2 m + 2 n + 1}{2}}}{2 m + 2 n + 1}

\int t^{\frac{4 n - 1}{2}} d t = \frac{2 t ^{\frac{4 n + 1}{2}}}{4 n + 1}

= \frac{2 t ^{\frac{4 m + 1}{2}}}{4 m + 1} - \frac{4 t ^{\frac{2 m + 2 n + 1}{2}}}{2 m + 2 n + 1} + \frac{2 t ^{\frac{4 n + 1}{2}}}{4 n + 1}

Füge eine Konstante zur Lösung hinzu

= \frac{2 t ^{\frac{4 m + 1}{2}}}{4 m + 1} - \frac{4 t ^{\frac{2 m + 2 n + 1}{2}}}{2 m + 2 n + 1} + \frac{2 t ^{\frac{4 n + 1}{2}}}{4 n + 1} + C

Graph

Beliebte Beispiele

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