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Beliebt Rechnen >

integral of (x^5+3x)/(x^4+1)

Lösung

\int \frac{x ^{5} + 3 x}{x ^{4} + 1} d x

Lösung

\frac{x ^{2}}{2} + arctan (x^{2}) + C

Schritte zur Lösung

\int \frac{x ^{5} + 3 x}{x ^{4} + 1} d x

Multipliziere aus

\frac{x ^{5} + 3 x}{x ^{4} + 1} : \frac{x ^{5}}{x ^{4} + 1} + \frac{3 x}{x ^{4} + 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 ^{5}}{x ^{4} + 1} d x + \int \frac{3 x}{x ^{4} + 1} d x

\int \frac{x ^{5}}{x ^{4} + 1} d x = \frac{x ^{2}}{2} - \frac{1}{2} arctan (x^{2})

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

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

Addiere gleiche Elemente:

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

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

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + arctan (x^{2}) + C

Graph

Beliebte Beispiele

integral of (3x)/((x+5)(2x-1))

\int \frac{3 x}{( x + 5 ) ( 2 x - 1 )} d x

integral of (-(csc(x)cot(x))/6)

\int (- \frac{csc ( x ) cot ( x )}{6}) d x

integral of (x^2+7)/x

\int \frac{x ^{2} + 7}{x} d x

integral of x*10^x

\int x \cdot 1 0^{x} d x

integral of e^{2t}arctan(e^t)

\int e^{2 t} arctan (e^{t}) d t