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Beliebt Rechnen >

integral of (x^3-2x+3)ln(x)

Lösung

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

Lösung

\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} - x^{2} ln (x) + \frac{x ^{2}}{2} + 3 x ln (x) - 3 x + C

Schritte zur Lösung

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

Multipliziere aus

(x^{3} - 2 x + 3) ln (x) : x^{3} ln (x) - 2 x ln (x) + 3 ln (x)

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int x^{3} ln (x) d x = \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}

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

\int 3 ln (x) d x = 3 (x ln (x) - x)

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} - (x^{2} ln (x) - \frac{x ^{2}}{2}) + 3 (x ln (x) - x)

Vereinfache

\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} - (x^{2} ln (x) - \frac{x ^{2}}{2}) + 3 (x ln (x) - x) : \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} - x^{2} ln (x) + \frac{x ^{2}}{2} + 3 x ln (x) - 3 x

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} - x^{2} ln (x) + \frac{x ^{2}}{2} + 3 x ln (x) - 3 x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} - x^{2} ln (x) + \frac{x ^{2}}{2} + 3 x ln (x) - 3 x + C

Graph

Beliebte Beispiele

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

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

integral of sqrt(6x^5)

\int 6 x^{5} d x

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

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

integral of (x^2+16x)/((x+8)^2)

\int \frac{x ^{2} + 16 x}{( x + 8 ) ^{2}} d x

integral of e^{3x-6}

\int e^{3 x - 6} d x