integral of 10xln(x+5)

Lösung

\int 10 x ln (x + 5) d x

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

Schritte zur Lösung

\int 10 x ln (x + 5) d x

Entferne die Konstante:

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

= 10 \cdot \int x ln (x + 5) d x

Wende U-Substitution an

= 10 \cdot \int (u - 5) ln (u) d u

Wende die partielle Integration an

= 10 (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)

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

Setze in

u = x + 5

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{x}{8} d x

y^{^{'}} = y^{2}, y (0) = 1

t an g e n t y = x^{4} + 9 x^{2} - x, (1, 9)

e^{x + y} - y^{^{'}} = 0

\frac{d y}{d x} = (x - 2 y cot (2 x))