integral from 1 to 3 of sqrt(1+(2x\sqrt{x^2-1))^2}

Lösung

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

\frac{52 + 12 2}{3} - 42 - 2

Dezimale

15.33333 \dots

Schritte zur Lösung

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

Faktorisiere

1 + (2 x x^{2} - 1)^{2} : (2 x + 1)^{2} (2 x - 1)^{2}

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

Wende U-Substitution an

= \int_{2 + 1}^{2 \cdot 3 + 1} \frac{u ( u - 1 )}{2} - \frac{u}{2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{2 + 1}^{2 \cdot 3 + 1} \frac{u ( u - 1 )}{2} d u - \int_{2 + 1}^{2 \cdot 3 + 1} \frac{u}{2} d u

\int_{2 + 1}^{2 \cdot 3 + 1} \frac{u ( u - 1 )}{2} d u = \frac{52 + 12 2}{3}

\int_{2 + 1}^{2 \cdot 3 + 1} \frac{u}{2} d u = 42 + 2

= \frac{52 + 12 2}{3} - (42 + 2)

Vereinfache

= \frac{52 + 12 2}{3} - 42 - 2

\int_{0}^{l} w cos (\frac{π}{2 l} x) d x

\int_{1}^{2} x lo g_{5} (x) d x

\int_{- 1}^{0} \frac{1}{e ^{x} + 1} d x

\int_{- 1}^{1} (x^{5} - x^{7}) d x

\int_{0}^{3} 4 t d t