integral of (y^2)/(y^2+1)

Lösung

\int \frac{y ^{2}}{y ^{2} + 1} d y

- arctan (y) + y + C

Schritte zur Lösung

\int \frac{y ^{2}}{y ^{2} + 1} d y

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

= \int - \frac{1}{y ^{2} + 1} + 1 d y

Wende die Summenregel an:

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

= - \int \frac{1}{y ^{2} + 1} d y + \int 1 d y

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

\int 1 d y = y

= - arctan (y) + y

Füge eine Konstante zur Lösung hinzu

= - arctan (y) + y + C

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

\int (4 x + 1) (2 x^{2} + x - 3)^{4} d x

\int \frac{( tan ( x ) ) ^{2}}{( cos ( x ) ) ^{2}} d x

\int 2 sin (\frac{x}{2}) cos (\frac{x}{2}) d x

\int \frac{1}{3} x ln (- 3 x) d x