integral of (x^2)/(x^2+25)

Lösung

\int \frac{x ^{2}}{x ^{2} + 25} d x

- 5 arctan (\frac{x}{5}) + x + C

Schritte zur Lösung

\int \frac{x ^{2}}{x ^{2} + 25} d x

\frac{x ^{2}}{x ^{2} + 25} = - \frac{25}{x ^{2} + 25} + 1

= \int - \frac{25}{x ^{2} + 25} + 1 d x

Wende integrale Substitution an

= \int 5 (- \frac{1}{u ^{2} + 1} + 1) d u

Entferne die Konstante:

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

= 5 \cdot \int - \frac{1}{u ^{2} + 1} + 1 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

= 5 (- \int \frac{1}{u ^{2} + 1} d u + \int 1 d u)

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

\int 1 d u = u

= 5 (- arctan (u) + u)

Setze in

u = \frac{x}{5}

ein

= 5 (- arctan (\frac{x}{5}) + \frac{x}{5})

Vereinfache

5 (- arctan (\frac{x}{5}) + \frac{x}{5}) : - 5 arctan (\frac{x}{5}) + x

= - 5 arctan (\frac{x}{5}) + x

Füge eine Konstante zur Lösung hinzu

= - 5 arctan (\frac{x}{5}) + x + C

\frac{\partial}{\partial x} (x y^{2})

\frac{\partial}{\partial x} (\frac{x + y}{e ^{z}})

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

\int sin^{4} (x co) s^{4} x d x

a re a y = \frac{1}{20 x ^{2}}, y = x, y = \frac{x}{16}