integral of 2(2x-1)ln(x-1)

Lösung

\int 2 (2 x - 1) ln (x - 1) d x

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

Schritte zur Lösung

\int 2 (2 x - 1) ln (x - 1) d x

Entferne die Konstante:

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

= 2 \cdot \int (2 x - 1) ln (x - 1) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

= 2 x^{2} ln (x - 1) - 2 x ln (x - 1) - x^{2}

Füge eine Konstante zur Lösung hinzu

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

\int (5 x^{3} + 10 x)^{3} (3 x^{2} + 2) d x

d er i v a t i v e x^{5} + 5^{x}

\int_{0}^{1} sin (x) d x

x \to 0 lim (\frac{2 x ^{2}}{sin ^{2} ( 8 x )})

t an g e n t f (x) = x^{4} - 2 x^{3} - 5 x - 3, at x = 1