integral from x to x^3 of y/x

Lösung

\int_{x}^{x^{3}} \frac{y}{x} d y

\frac{x ( x ^{4} - 1 )}{2}

Schritte zur Lösung

\int_{x}^{x^{3}} \frac{y}{x} d y

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{x} \cdot \int_{x}^{x^{3}} y d y

Wende die Potenzregel an

= \frac{1}{x} [\frac{y ^{2}}{2}]_{x}^{x^{3}}

Vereinfache

\frac{1}{x} [\frac{y ^{2}}{2}]_{x}^{x^{3}} : \frac{[ \frac{y ^{2}}{2} ] _{x}^{x^{3}}}{x}

= \frac{[ \frac{y ^{2}}{2} ] _{x}^{x^{3}}}{x}

Berechne die Grenzen:

\frac{x ^{6}}{2} - \frac{x ^{2}}{2}

= \frac{\frac{x ^{6}}{2} - \frac{x ^{2}}{2}}{x}

Vereinfache

= \frac{x ( x ^{4} - 1 )}{2}

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