integral of (2x^2-1)cos(3x)

Lösung

\int (2 x^{2} - 1) cos (3 x) d x

\frac{2}{3} x^{2} sin (3 x) + \frac{4}{9} x cos (3 x) - \frac{13}{27} sin (3 x) + C

Schritte zur Lösung

\int (2 x^{2} - 1) cos (3 x) d x

Wende die partielle Integration an

= \frac{1}{3} sin (3 x) (2 x^{2} - 1) - \int \frac{4}{3} x sin (3 x) d x

\int \frac{4}{3} x sin (3 x) d x = \frac{4}{3} (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x))

= \frac{1}{3} sin (3 x) (2 x^{2} - 1) - \frac{4}{3} (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x))

Vereinfache

\frac{1}{3} sin (3 x) (2 x^{2} - 1) - \frac{4}{3} (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) : \frac{2}{3} x^{2} sin (3 x) + \frac{4}{9} x cos (3 x) - \frac{13}{27} sin (3 x)

= \frac{2}{3} x^{2} sin (3 x) + \frac{4}{9} x cos (3 x) - \frac{13}{27} sin (3 x)

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} x^{2} sin (3 x) + \frac{4}{9} x cos (3 x) - \frac{13}{27} sin (3 x) + C

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