integral of xsin^4(x)

Lösung

\int x sin^{4} (x) d x

- \frac{1}{4} x sin^{3} (x) cos (x) + \frac{3}{8} x^{2} - \frac{3}{16} x sin (2 x) + \frac{1}{16} sin^{4} (x) - \frac{3 x ^{2}}{16} - \frac{3}{32} cos (2 x) + C

Schritte zur Lösung

\int x sin^{4} (x) d x

Wende die partielle Integration an

= x (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - \int - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) d x

\int - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) d x = - \frac{1}{16} sin^{4} (x) + \frac{3}{8} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))

= x (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - (- \frac{1}{16} sin^{4} (x) + \frac{3}{8} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)))

Vereinfache

x (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - (- \frac{1}{16} sin^{4} (x) + \frac{3}{8} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))) : - \frac{1}{4} x sin^{3} (x) cos (x) + \frac{3}{8} x^{2} - \frac{3}{16} x sin (2 x) + \frac{1}{16} sin^{4} (x) - \frac{3 x ^{2}}{16} - \frac{3}{32} cos (2 x)

= - \frac{1}{4} x sin^{3} (x) cos (x) + \frac{3}{8} x^{2} - \frac{3}{16} x sin (2 x) + \frac{1}{16} sin^{4} (x) - \frac{3 x ^{2}}{16} - \frac{3}{32} cos (2 x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} x sin^{3} (x) cos (x) + \frac{3}{8} x^{2} - \frac{3}{16} x sin (2 x) + \frac{1}{16} sin^{4} (x) - \frac{3 x ^{2}}{16} - \frac{3}{32} cos (2 x) + C

\int \frac{1 + x ^{2}}{x ^{4}} d x

\int ((x - 5)^{2} - x)^{2} d x

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

\int \frac{( x ^{3} )}{x ^{2} + 1} d x

\int t sec^{2} (t) d t