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

Lösung

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

\frac{1}{2} (- arctan (x) + x) + C

Schritte zur Lösung

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

\frac{x ^{2}}{2 + 2 x ^{2}} = \frac{1}{2} \frac{x ^{2}}{1 + x ^{2}}

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

Entferne die Konstante:

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

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

\frac{x ^{2}}{1 + x ^{2}} = - \frac{1}{1 + x ^{2}} + 1

= \frac{1}{2} \cdot \int - \frac{1}{1 + 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

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

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

\int 1 d x = x

= \frac{1}{2} (- arctan (x) + x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (- arctan (x) + x) + C

\int 2 x^{2} (x^{3} + 8)^{5} d x

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

\int \frac{e ^{6 x}}{1 + e ^{6 x}} d x

\int z e^{yz} d y

\int 12 x^{2} (4 x^{3} - 3) d x