integral of (2x+5)/(x(x^2+1))

Lösung

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

2 arctan (x) + 5 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x + 5}{x ( x ^{2} + 1 )} : \frac{2 x}{x ( x ^{2} + 1 )} + \frac{5}{x ( 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{2 x}{x ( x ^{2} + 1 )} d x + \int \frac{5}{x ( x ^{2} + 1 )} d x

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

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

= 2 arctan (x) + 5 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Füge eine Konstante zur Lösung hinzu

= 2 arctan (x) + 5 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

\int \frac{x}{( x - 1 ) ^{4}} d x

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