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Beliebt Rechnen >

integral of sqrt(4t^2+9t^4)

Lösung

\int 4 t^{2} + 9 t^{4} d t

Lösung

\frac{1}{27} (4 + 9 t^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int 4 t^{2} + 9 t^{4} d t

Faktorisiere

4 t^{2} + 9 t^{4} : t^{2} 4 + 9 t^{2}

= \int t^{2} 4 + 9 t^{2} d t

t^{2} = (t),

angenommen

t \geq 0

= \int t 4 + 9 t^{2} d t

Wende U-Substitution an

= \int \frac{u}{18} d u

Entferne die Konstante:

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

= \frac{1}{18} \cdot \int u d u

Wende die Potenzregel an

= \frac{1}{18} \cdot \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = 4 + 9 t^{2}

ein

= \frac{1}{18} \cdot \frac{2}{3} (4 + 9 t^{2})^{\frac{3}{2}}

Vereinfache

\frac{1}{18} \cdot \frac{2}{3} (4 + 9 t^{2})^{\frac{3}{2}} : \frac{1}{27} (4 + 9 t^{2})^{\frac{3}{2}}

= \frac{1}{27} (4 + 9 t^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{27} (4 + 9 t^{2})^{\frac{3}{2}} + C

Graph

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