integral from 0 to 2 of 3xsqrt(x^2+4a^2)

Lösung

\int_{0}^{2} 3 x x^{2} + 4 a^{2} d x

8 ((a^{2} + 1)^{\frac{3}{2}} - a^{3})

Schritte zur Lösung

\int_{0}^{2} 3 x x^{2} + 4 a^{2} d x

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{2} x x^{2} + 4 a^{2} d x

Wende U-Substitution an

= 3 \cdot \int_{4 a^{2}}^{4 + 4 a^{2}} \frac{u}{2} d u

Entferne die Konstante:

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

= 3 \cdot \frac{1}{2} \cdot \int_{4 a^{2}}^{4 + 4 a^{2}} u d u

Wende die Potenzregel an

= 3 \cdot \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

Vereinfache

3 \cdot \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}} : \frac{3}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

= \frac{3}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{4 a^{2}}^{4 + 4 a^{2}}

Berechne die Grenzen:

\frac{16}{3} (1 + a^{2})^{\frac{3}{2}} - \frac{16}{3} a^{3}

= \frac{3}{2} (\frac{16}{3} (1 + a^{2})^{\frac{3}{2}} - \frac{16}{3} a^{3})

Vereinfache

= 8 ((a^{2} + 1)^{\frac{3}{2}} - a^{3})

\int_{- 2}^{2} 1 - x^{2} d y

\int_{0}^{1} \frac{1}{2} x^{3} + 2 d x

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

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