integral of t^2cos^2(t)

Lösung

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

\frac{1}{3} t^{3} cos^{2} (t) - \frac{1}{6} t^{3} cos (2 t) + \frac{1}{4} t^{2} sin (2 t) + \frac{1}{4} t cos (2 t) - \frac{1}{8} sin (2 t) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{1}{3} t^{3} cos^{2} (t) - \int - \frac{1}{3} t^{3} sin (2 t) d t

\int - \frac{1}{3} t^{3} sin (2 t) d t = \frac{1}{6} t^{3} cos (2 t) - \frac{1}{8} (2 t^{2} sin (2 t) + 2 t cos (2 t) - sin (2 t))

= \frac{1}{3} t^{3} cos^{2} (t) - (\frac{1}{6} t^{3} cos (2 t) - \frac{1}{8} (2 t^{2} sin (2 t) + 2 t cos (2 t) - sin (2 t)))

Vereinfache

\frac{1}{3} t^{3} cos^{2} (t) - (\frac{1}{6} t^{3} cos (2 t) - \frac{1}{8} (2 t^{2} sin (2 t) + 2 t cos (2 t) - sin (2 t))) : \frac{1}{3} t^{3} cos^{2} (t) - \frac{1}{6} t^{3} cos (2 t) + \frac{1}{4} t^{2} sin (2 t) + \frac{1}{4} t cos (2 t) - \frac{1}{8} sin (2 t)

= \frac{1}{3} t^{3} cos^{2} (t) - \frac{1}{6} t^{3} cos (2 t) + \frac{1}{4} t^{2} sin (2 t) + \frac{1}{4} t cos (2 t) - \frac{1}{8} sin (2 t)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} t^{3} cos^{2} (t) - \frac{1}{6} t^{3} cos (2 t) + \frac{1}{4} t^{2} sin (2 t) + \frac{1}{4} t cos (2 t) - \frac{1}{8} sin (2 t) + C

\int \frac{x + 10}{2 - x - 3 x ^{2}} d x

\frac{d}{d x} (3 x^{2} - 9)

n = 0 \sum \infty \frac{4 ^{n}}{5 ^{n}}

d er i v a t i v e f (x) = (3 x + 70)^{50}

y^{^{''}} + 6.2832 y = 0