integral of 4x^3+3x^2+2x+1+1/x+1/(x^3)

Lösung

\int 4 x^{3} + 3 x^{2} + 2 x + 1 + \frac{1}{x} + \frac{1}{x ^{3}} d x

x^{4} + x^{3} + x^{2} + x + ln ∣ x ∣ - \frac{1}{2 x ^{2}} + C

Schritte zur Lösung

\int 4 x^{3} + 3 x^{2} + 2 x + 1 + \frac{1}{x} + \frac{1}{x ^{3}} 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 4 x^{3} d x + \int 3 x^{2} d x + \int 2 x d x + \int 1 d x + \int \frac{1}{x} d x + \int \frac{1}{x ^{3}} d x

\int 4 x^{3} d x = x^{4}

\int 3 x^{2} d x = x^{3}

\int 2 x d x = x^{2}

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= x^{4} + x^{3} + x^{2} + x + ln ∣ x ∣ - \frac{1}{2 x ^{2}}

Füge eine Konstante zur Lösung hinzu

= x^{4} + x^{3} + x^{2} + x + ln ∣ x ∣ - \frac{1}{2 x ^{2}} + C