integral of xln(ax+b)

Lösung

\int x ln (a x + b) d x

\frac{1}{2} x^{2} ln (a x + b) - \frac{( a x + b ) ^{2}}{4 a ^{2}} + \frac{b x}{a} + \frac{b ^{2}}{a ^{2}} - b^{2} \frac{1}{2 a ^{2}} ln ∣ a x + b ∣ + C

Schritte zur Lösung

\int x ln (a x + b) d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} ln (a x + b) - \int \frac{a x ^{2}}{2 ( a x + b )} d x

\int \frac{a x ^{2}}{2 ( a x + b )} d x = \frac{( a x + b ) ^{2}}{4 a ^{2}} - \frac{b x}{a} - \frac{b ^{2}}{a ^{2}} + b^{2} \frac{1}{2 a ^{2}} ln ∣ a x + b ∣

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} ln (a x + b) - \frac{( a x + b ) ^{2}}{4 a ^{2}} + \frac{b x}{a} + \frac{b ^{2}}{a ^{2}} - b^{2} \frac{1}{2 a ^{2}} ln ∣ a x + b ∣ + C

\int \frac{cos ( 3 x )}{5 + 2 sin ( 3 x )} d x

\int \frac{1}{1 - cos ( 2 x )} d x

\int \frac{1}{( 2 x - 3 ) 5 - 12 x + 4 x ^{2}} d x

\int (t - \frac{2}{t}) (t + \frac{2}{t}) d t

\int \frac{x}{( 9 + x ^{2} ) ^{3}} d x