de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of ln(sqrt(x)+1)

Lösung

\int ln (x + 1) d x

Lösung

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

Schritte zur Lösung

\int ln (x + 1) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (2ln|x|)/x

\int \frac{2 ln ∣ x ∣}{x} d x

integral of (cos(7sqrt(x)))/(sqrt(x))

\int \frac{cos ( 7 x )}{x} d x

integral of x^9sin(x^{10})

\int x^{9} sin (x^{10}) d x

integral of (-3x)/((x+1)-sqrt(x+1))

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

integral of (2+3^x)

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