integral of (5-x^2)/(3x^3+x)

Lösung

\int \frac{5 - x ^{2}}{3 x ^{3} + x} d x

5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1 + C

Schritte zur Lösung

\int \frac{5 - x ^{2}}{3 x ^{3} + x} d x

Multipliziere aus

\frac{5 - x ^{2}}{3 x ^{3} + x} : \frac{5}{3 x ^{3} + x} - \frac{x ^{2}}{3 x ^{3} + 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 \frac{5}{3 x ^{3} + x} d x - \int \frac{x ^{2}}{3 x ^{3} + x} d x

\int \frac{5}{3 x ^{3} + x} d x = 5 (ln ∣ x ∣ - \frac{1}{2} ln 3 x^{2} + 1)

\int \frac{x ^{2}}{3 x ^{3} + x} d x = \frac{1}{6} ln 3 x^{2} + 1

= 5 (ln ∣ x ∣ - \frac{1}{2} ln 3 x^{2} + 1) - \frac{1}{6} ln 3 x^{2} + 1

Vereinfache

5 (ln ∣ x ∣ - \frac{1}{2} ln 3 x^{2} + 1) - \frac{1}{6} ln 3 x^{2} + 1 : 5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1

= 5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= 5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1 + C

f (x) = x e^{3 x}

\frac{\partial}{\partial x} (tan (6 + 2 x^{2} y^{4} z^{2}))

\int_{2}^{5} x d x

x \to 11 lim (x^{2})

\frac{d}{d t} (sec (t))