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Beliebt Rechnen >

integral of x*ln(5+x)

Lösung

\int x \cdot ln (5 + x) d x

Lösung

ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75) + C

Schritte zur Lösung

\int x ln (5 + x) d x

Wende U-Substitution an

= \int (u - 5) ln (u) d u

Wende die partielle Integration an

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \int \frac{1}{2} (u - 10) d u

\int \frac{1}{2} (u - 10) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

Setze in

u = 5 + x

ein

= ln (5 + x) (\frac{( 5 + x ) ^{2}}{2} - 5 (5 + x)) - \frac{1}{2} (\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x))

Vereinfache

ln (5 + x) (\frac{( 5 + x ) ^{2}}{2} - 5 (5 + x)) - \frac{1}{2} (\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x)) : ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75)

= ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75)

Füge eine Konstante zur Lösung hinzu

= ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75) + C

Graph

Beliebte Beispiele

integral of pi/2-5x^{-0.5}

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fläche y^2=x,x-2y=3

a re a y^{2} = x, x - 2 y = 3