integral from 0 to 1 of 6(y/x)^4

Lösung

\int_{0}^{1} 6 (\frac{y}{x})^{4} d y

\frac{6}{5 x ^{4}}

Schritte zur Lösung

\int_{0}^{1} 6 (\frac{y}{x})^{4} d y

Entferne die Konstante:

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

= 6 \cdot \int_{0}^{1} (\frac{y}{x})^{4} d y

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= 6 \cdot \int_{0}^{1} \frac{y ^{4}}{x ^{4}} d y

Entferne die Konstante:

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

= 6 \cdot \frac{1}{x ^{4}} \cdot \int_{0}^{1} y^{4} d y

Wende die Potenzregel an

= 6 \cdot \frac{1}{x ^{4}} [\frac{y ^{5}}{5}]_{0}^{1}

Vereinfache

6 \cdot \frac{1}{x ^{4}} [\frac{y ^{5}}{5}]_{0}^{1} : \frac{6 [ \frac{y ^{5}}{5} ] _{0}^{1}}{x ^{4}}

= \frac{6 [ \frac{y ^{5}}{5} ] _{0}^{1}}{x ^{4}}

Berechne die Grenzen:

\frac{1}{5}

= \frac{6 \cdot \frac{1}{5}}{x ^{4}}

Vereinfache

= \frac{6}{5 x ^{4}}

\int_{1}^{25} \frac{1}{1 + x} d x

\int_{1}^{\infty} \frac{1}{n ^{3}}

\int_{1}^{\infty} \frac{1}{n ^{4}}

\int_{- 2}^{5} (10 x - x^{2}) d x

\int_{1}^{2} (3 x + 2) ln (x) d x