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

Lösung

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

2 ln ∣ x - 1 ∣ - \frac{3}{x - 1} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

2 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - \frac{1}{x - 1} : 2 ln ∣ x - 1 ∣ - \frac{3}{x - 1}

= 2 ln ∣ x - 1 ∣ - \frac{3}{x - 1}

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x - 1 ∣ - \frac{3}{x - 1} + C

\int (3 x^{2} + 4 x + e^{x} + sec (x) tan (x)) d x

\int i sin (x) d x

\int sin (x + 4) d x

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

\int - 0.8 x + 40 d x