integral from x^3 to x^6 of (2t-1)^3

Lösung

\int_{x^{3}}^{x^{6}} (2 t - 1)^{3} d t

x^{3} (2 x^{21} - 4 x^{15} + x^{9} + 4 x^{6} - 4 x^{3} + 1)

Schritte zur Lösung

\int_{x^{3}}^{x^{6}} (2 t - 1)^{3} d t

Wende U-Substitution an

= \int_{2 x^{3} - 1}^{2 x^{6} - 1} \frac{u ^{3}}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{2 x^{3} - 1}^{2 x^{6} - 1} u^{3} d u

Wende die Potenzregel an

= \frac{1}{2} [\frac{u ^{4}}{4}]_{2 x^{3} - 1}^{2 x^{6} - 1}

Berechne die Grenzen:

\frac{( 2 x ^{6} - 1 ) ^{4}}{4} - \frac{( 2 x ^{3} - 1 ) ^{4}}{4}

= \frac{1}{2} (\frac{( 2 x ^{6} - 1 ) ^{4}}{4} - \frac{( 2 x ^{3} - 1 ) ^{4}}{4})

Vereinfache

= x^{3} (2 x^{21} - 4 x^{15} + x^{9} + 4 x^{6} - 4 x^{3} + 1)