integral from 0 to x^2 of y/(x^5+1)

Lösung

\int_{0}^{x^{2}} \frac{y}{x ^{5} + 1} d y

\frac{x ^{4}}{2 ( 1 + x ^{5} )}

Schritte zur Lösung

\int_{0}^{x^{2}} \frac{y}{x ^{5} + 1} d y

Entferne die Konstante:

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

= \frac{1}{1 + x ^{5}} \cdot \int_{0}^{x^{2}} y d y

Wende die Potenzregel an

= \frac{1}{1 + x ^{5}} [\frac{y ^{2}}{2}]_{0}^{x^{2}}

Vereinfache

\frac{1}{1 + x ^{5}} [\frac{y ^{2}}{2}]_{0}^{x^{2}} : \frac{[ \frac{y ^{2}}{2} ] _{0}^{x^{2}}}{1 + x ^{5}}

= \frac{[ \frac{y ^{2}}{2} ] _{0}^{x^{2}}}{1 + x ^{5}}

Berechne die Grenzen:

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

= \frac{\frac{x ^{4}}{2}}{1 + x ^{5}}

Vereinfache

= \frac{x ^{4}}{2 ( 1 + x ^{5} )}

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