integral of (3t+t^2)sin(3t)

解答

\int (3 t + t^{2}) sin (3 t) d t

- t cos (3 t) + \frac{1}{3} sin (3 t) - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t)) + C

求解步骤

\int (3 t + t^{2}) sin (3 t) d t

乘开

(3 t + t^{2}) sin (3 t) : 3 t sin (3 t) + t^{2} sin (3 t)

= \int 3 t sin (3 t) + t^{2} sin (3 t) d t

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3 t sin (3 t) d t + \int t^{2} sin (3 t) d t

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

\int t^{2} sin (3 t) d t = - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t))

= - t cos (3 t) + \frac{1}{3} sin (3 t) - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t))

解答补常数

= - t cos (3 t) + \frac{1}{3} sin (3 t) - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t)) + C

\int ln (1 + t^{2}) d t

d er i v a t i v e 2 w^{3 π}, at 7

in v erse l a pl a ce \frac{k}{s}

\frac{d}{d x} (cos^{2} (9 x) + sin^{2} (9 x))

\frac{d y}{d x} + \frac{y}{x ^{2} + x} = x + 1