integral of ((2x+3))/((x+1)^2)

Lösung

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

2 ln ∣ x + 1 ∣ - \frac{1}{x + 1} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x + 3}{( x + 1 ) ^{2}} : \frac{2 x}{( x + 1 ) ^{2}} + \frac{3}{( 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{3}{( x + 1 ) ^{2}} d x

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

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

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

Vereinfache

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

= 2 ln ∣ x + 1 ∣ - \frac{1}{x + 1}

Füge eine Konstante zur Lösung hinzu

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

t an g e n t 110 x - 16 x^{2}

\frac{d}{d x} (16)

\int \frac{7 - 6 x ^{3} + 4 x ^{6}}{x ^{6}} d x

\frac{d}{d x} (arctan (\frac{1}{3 x}))

\frac{d}{d r} (\frac{r}{r ^{2} + 5})