integral from 0 to 2 of 4x^2e^{2x}

Lösung

\int_{0}^{2} 4 x^{2} e^{2 x} d x

5 e^{4} - 1

Dezimale

271.99075 \dots

Schritte zur Lösung

\int_{0}^{2} 4 x^{2} e^{2 x} d x

Entferne die Konstante:

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

= 4 \cdot \int_{0}^{2} x^{2} e^{2 x} d x

Wende die partielle Integration an

= 4 [\frac{1}{2} e^{2 x} x^{2} - \int e^{2 x} x d x]_{0}^{2}

\int e^{2 x} x d x = \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

= 4 [\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})]_{0}^{2}

Berechne die Grenzen:

2 e^{4} - \frac{3 e ^{4}}{4} - \frac{1}{4}

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

Vereinfache

= 5 e^{4} - 1

\int_{1}^{2} (3 x^{2} - 5 x) d x

\int_{- 1}^{1} 4^{x} d x

\int_{a}^{b} e^{- t^{2}} d t

\int_{- 5}^{5} (9 π z + \frac{81 π}{2}) d z

\int_{0}^{6} 16 - 2 x - 2 d x