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Beliebt Rechnen >

integral of sqrt(1+(2x(x^2+1)^{1/2))^2}

Lösung

\int 1 + (2 x (x^{2} + 1)^{\frac{1}{2}})^{2} d x

Lösung

\frac{2 x ^{3}}{3} + x + C

Schritte zur Lösung

\int 1 + (2 x (x^{2} + 1)^{\frac{1}{2}})^{2} d x

Vereinfache

1 + (2 x (x^{2} + 1)^{\frac{1}{2}})^{2} : 1 + 4 x^{2} (x^{2} + 1)

= \int 1 + 4 x^{2} (x^{2} + 1) d x

Multipliziere aus

4 x^{2} (x^{2} + 1) : 4 x^{4} + 4 x^{2}

= \int 1 + 4 x^{4} + 4 x^{2} d x

Faktorisiere

4 x^{4} + 4 x^{2} + 1 : (2 x^{2} + 1)^{2}

= \int (2 x^{2} + 1)^{2} d x

(2 x^{2} + 1)^{2} = ((2 x^{2} + 1)),

angenommen

(2 x^{2} + 1) \geq 0

= \int 2 x^{2} + 1 d x

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 x^{2} d x + \int 1 d x

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

\int 1 d x = x

= \frac{2 x ^{3}}{3} + x

Füge eine Konstante zur Lösung hinzu

= \frac{2 x ^{3}}{3} + x + C

Graph

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\int (4 + x^{2})^{\frac{3}{2}} d x

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\int \frac{( x ^{4} - 1 )}{x ^{2} ( x + 1 ) ^{2}} d x