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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

Wende die Summenregel an:

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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