integral of (x^3+x+1)/(x^3+x)

Lösung

\int \frac{x ^{3} + x + 1}{x ^{3} + x} d x

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

Schritte zur Lösung

\int \frac{x ^{3} + x + 1}{x ^{3} + x} d x

\frac{x ^{3} + x + 1}{x ^{3} + x} = \frac{1}{x ^{3} + x} + 1

= \int \frac{1}{x ^{3} + x} + 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

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

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

\int 1 d x = x

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

Füge eine Konstante zur Lösung hinzu

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

\int cos (θ) tan (θ) d θ

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

\int 10 e^{t} - t e^{t} + 1 d t

\int (1 + x) \cdot x d x

\int 5 e^{x} - 6 sin (x) d x