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

Lösung

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

\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 10 x - 31 ln ∣ x + 3 ∣ + 3 + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{3} + x - 1}{x + 3} : \frac{x ^{3}}{x + 3} + \frac{x}{x + 3} - \frac{1}{x + 3}

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

\int \frac{x ^{3}}{x + 3} d x = \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣

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

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

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣ + x + 3 - 3 ln ∣ x + 3 ∣ - ln ∣ x + 3 ∣

Vereinfache

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 10 x - 31 ln ∣ x + 3 ∣ + 3

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 10 x - 31 ln ∣ x + 3 ∣ + 3 + C

\int csc^{5} (x) d x

d er i v a t i v e \frac{4 x}{( x ^{3} - 3 ) ^{2}}

d er i v a t i v e x + tan (x)

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

\int x \frac{1}{1} d x