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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{3}}{3} + \frac{3}{2} x^{2} - x - ln x^{2} + 1 + 2 arctan (x) + \frac{3}{2} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{4} + 3 x ^{3} + x + 1}{x ^{2} + 1} : \frac{x ^{4}}{x ^{2} + 1} + \frac{3 x ^{3}}{x ^{2} + 1} + \frac{x}{x ^{2} + 1} + \frac{1}{x ^{2} + 1}

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

\int \frac{x ^{4}}{x ^{2} + 1} d x = \frac{x ^{3}}{3} - x + arctan (x)

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

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

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

= \frac{x ^{3}}{3} - x + arctan (x) + \frac{3}{2} (x^{2} + 1 - ln x^{2} + 1) + \frac{1}{2} ln x^{2} + 1 + arctan (x)

Vereinfache

\frac{x ^{3}}{3} - x + arctan (x) + \frac{3}{2} (x^{2} + 1 - ln x^{2} + 1) + \frac{1}{2} ln x^{2} + 1 + arctan (x) : \frac{x ^{3}}{3} + \frac{3}{2} x^{2} - x - ln x^{2} + 1 + 2 arctan (x) + \frac{3}{2}

= \frac{x ^{3}}{3} + \frac{3}{2} x^{2} - x - ln x^{2} + 1 + 2 arctan (x) + \frac{3}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{3}{2} x^{2} - x - ln x^{2} + 1 + 2 arctan (x) + \frac{3}{2} + C

Graph

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