integral from x to x^2 of (y/x)

Lösung

\int_{x}^{x^{2}} (\frac{y}{x}) d y

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

Schritte zur Lösung

\int_{x}^{x^{2}} \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^{2}} y d y

Wende die Potenzregel an

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{x ^{4}}{2} - \frac{x ^{2}}{2}

= \frac{\frac{x ^{4}}{2} - \frac{x ^{2}}{2}}{x}

Vereinfache

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

\int_{0}^{6} \frac{∣ 2 x - 5 ∣}{x + 1} d x

\int_{\frac{1}{2}}^{2} (2 - \frac{2}{x ^{2}}) d x

\int_{0}^{6} x d x

\int_{- 1}^{1} (2 x^{2} + 1) d x

\int_{1}^{2} \frac{x ^{2} + 3}{x} d x