de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

- ln x^{2} + 1 + 2 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

= - ln x^{2} + 1 + 2 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - ln x^{2} + 1 + 2 ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

integral of (ln(arctan(x)))/(1+x^2)

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

integral of 1/(4x^3+4x^2+x)

\int \frac{1}{4 x ^{3} + 4 x ^{2} + x} d x

integral of 1/(3cos(x))

\int \frac{1}{3 cos ( x )} d x

integral of 2/5 x^{5/3}+C

\int \frac{2}{5} x^{\frac{5}{3}} + C d x

integral of (x^2cos^2(x))

\int (x^{2} cos^{2} (x)) d x