integral of xsqrt(x^4-1)

Lösung

\int x x^{4} - 1 d x

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

Schritte zur Lösung

\int x x^{4} - 1 d x

Wende U-Substitution an

= \int \frac{u - 1}{4 u} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{u - 1}{u} d u

Wende die partielle Integration an

= \frac{1}{4} (2 u u - 1 - \int \frac{u}{u - 1} d u)

\int \frac{u}{u - 1} d u = \frac{u}{u - 1} (u - 1) + ln u + u - 1

= \frac{1}{4} (2 u u - 1 - (\frac{u}{u - 1} (u - 1) + ln u + u - 1))

Setze in

u = x^{4}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (\frac{8 x ^{3} - 2 x ^{2} + 3}{2 x})

\int (y - x^{3}) d x

\int \frac{5 x - 4}{8 x - 5 ^{2} + 9} d x

\frac{d y}{d x} + y^{9} x + 4 y = 0

\int_{0}^{2} 4 x + 1 d x