integral of x^3arctan(sqrt(x^2-1))

Lösung

\int x^{3} arctan (x^{2} - 1) d x

\frac{1}{4} x^{4} arctan (x^{2} - 1) - \frac{1}{4} (\frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1) + C

Schritte zur Lösung

\int x^{3} arctan (x^{2} - 1) d x

Wende die partielle Integration an

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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