integral of xsin(2x)cos(x)

Lösung

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

- \frac{2}{3} x cos^{3} (x) + \frac{2}{3} (sin (x) - \frac{1}{3} sin^{3} (x)) + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

= - \frac{2}{3} x cos^{3} (x) - (- \frac{2}{3} (sin (x) - \frac{sin ^{3} ( x )}{3}))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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

\int 4 - x^{2} \cdot x d x

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

\int e^{- \frac{2 x}{3}} d x

\int \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + \frac{23}{6} d x