integral of (1+2x^2)^2

Lösung

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

x + \frac{4 x ^{3}}{3} + \frac{4 x ^{5}}{5} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int 1 d x = x

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

\int 4 x^{4} d x = \frac{4 x ^{5}}{5}

= x + \frac{4 x ^{3}}{3} + \frac{4 x ^{5}}{5}

Füge eine Konstante zur Lösung hinzu

= x + \frac{4 x ^{3}}{3} + \frac{4 x ^{5}}{5} + C

\frac{d y}{d x} = \frac{2 x}{1 + 2 y}, y (2) = 1

\frac{2 x}{y} d x - \frac{x ^{2}}{y ^{2}} d y = 0

d er i v a t i v e f (x) = \frac{2 x}{7} + 1

\int (\frac{3}{x ^{\frac{3}{4}}} - 4 3 x + 1) d x

\frac{d y}{d x} = (x - 2) e^{- 2 y}