integral of pi/2 x^2e^{pix}

Lösung

\int \frac{π}{2} x^{2} e^{π x} d x

\frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x} + C

Schritte zur Lösung

\int \frac{π}{2} x^{2} e^{π x} d x

Entferne die Konstante:

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

= \frac{π}{2} \cdot \int x^{2} e^{π x} d x

Wende die partielle Integration an

= \frac{π}{2} (\frac{1}{π} e^{π x} x^{2} - \int \frac{2}{π} e^{π x} x d x)

\int \frac{2}{π} e^{π x} x d x = \frac{2}{π} (\frac{1}{π} e^{π x} x - \frac{1}{π ^{2}} e^{π x})

= \frac{π}{2} (\frac{1}{π} e^{π x} x^{2} - \frac{2}{π} (\frac{1}{π} e^{π x} x - \frac{1}{π ^{2}} e^{π x}))

Vereinfache

\frac{π}{2} (\frac{1}{π} e^{π x} x^{2} - \frac{2}{π} (\frac{1}{π} e^{π x} x - \frac{1}{π ^{2}} e^{π x})) : \frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x}

= \frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x} + C

\int \frac{7 x ^{2} + 3 x + 8}{x ^{3} - 1} d x

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

\int \frac{x ^{2} + 12 x - 12}{x ^{3} - 4 x} d x

\int \frac{x ^{2} - 2 x + 4}{x ^{3} + 4 x} d x

\int tan^{3} (x) sin^{2} (x) d x