integral of 1/2 ln|x^2+1|

Lösung

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

\frac{1}{2} x ln x^{2} + 1 + arctan (x) - x + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

\int \frac{2 x ^{2}}{x ^{2} + 1} d x = 2 (- arctan (x) + x)

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int sin (3 x) sin (2 x) sin (x) d x

\frac{1}{4} y^{^{''}} + 4 y = 2 tan (4 t) - \frac{1}{3} e^{3 t}

\frac{d}{d x} (3 x - 8)

in v erse l a pl a ce \frac{2}{s ^{2} + 2 s + 10} \cdot \frac{1}{s}

\frac{d}{d x} (1 - (1 - \frac{x}{24})^{3})