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Beliebt Rechnen >

integral of xln(sqrt(x+2))

Lösung

\int x ln (x + 2) d x

Lösung

\frac{1}{8} (2 x^{2} ln (x + 2) - 8 ln (x + 2) - x^{2} + 4 x + 12) + C

Schritte zur Lösung

\int x ln (x + 2) d x

Vereinfache

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

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Wende die partielle Integration an

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

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

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

Setze in

u = x + 2

ein

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

Vereinfache

\frac{1}{2} (ln (x + 2) (\frac{( x + 2 ) ^{2}}{2} - 2 (x + 2)) - \frac{1}{2} (\frac{( x + 2 ) ^{2}}{2} - 4 (x + 2))) : \frac{1}{8} (2 x^{2} ln (x + 2) - 8 ln (x + 2) - x^{2} + 4 x + 12)

= \frac{1}{8} (2 x^{2} ln (x + 2) - 8 ln (x + 2) - x^{2} + 4 x + 12)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (2 x^{2} ln (x + 2) - 8 ln (x + 2) - x^{2} + 4 x + 12) + C

Graph

Beliebte Beispiele

integral of 1/(t^2-9)

\int \frac{1}{t ^{2} - 9} d t

(\partial)/(\partial x)((e^{-6x})cos(y))

\frac{\partial}{\partial x} ((e^{- 6 x}) cos (y))

integral of 1/(\sqrt[3]{x-1)}

\int \frac{1}{3 x - 1} d x

integral of (ye^{2xy}+x)

\int (y e^{2 x y} + x) d x

tangent y=x^3-5x+1,(3,13)

t an g e n t y = x^{3} - 5 x + 1, (3, 13)