integral of (x^2+1)sin(2x)

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int (sin (a) x - e^{\frac{x}{b}}) d x

in t e g r a l csc^{2} (x)

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

\int \frac{6}{4 - 3 x} d x

\int sin^{4} (6 x) cos (6 x) d x