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

Lösung

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

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

Schritte zur Lösung

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

\int x ln (x) d x = \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{1}{5 5 x ^{2}} d x

\int \frac{x ^{2} - 2}{x ^{2} - x - 2} d x

\int sec^{- 2} (x) tan^{3} (x) d x

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\int \frac{ln ( x ^{2} )}{x} d x