integral of-3.4(4.5-t)sin(0.4(3-t))+80

解答

\int - 3.4 (4.5 - t) sin (0.4 (3 - t)) + 80 d t

8.5 t cos (0.4 (- t + 3)) + 80 t - 38.25 cos (0.4 (- t + 3)) + 21.25 sin (0.4 (- t + 3)) - 240 + C

求解步骤

\int - 3.4 (4.5 - t) sin (0.4 (3 - t)) + 80 d t

使用换元积分法

= \int 3.4 u sin (0.4 u) + 5.1 sin (0.4 u) - 80 d u

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 3.4 u sin (0.4 u) d u + \int 5.1 sin (0.4 u) d u - \int 80 d u

\int 3.4 u sin (0.4 u) d u = 3.4 (- 2.5 u cos (0.4 u) + 6.25 sin (0.4 u))

\int 5.1 sin (0.4 u) d u = - 12.75 cos (0.4 u)

\int 80 d u = 80 u

= 3.4 (- 2.5 u cos (0.4 u) + 6.25 sin (0.4 u)) - 12.75 cos (0.4 u) - 80 u

u = 3 - t

代回

= 3.4 (- 2.5 (3 - t) cos (0.4 (3 - t)) + 6.25 sin (0.4 (3 - t))) - 12.75 cos (0.4 (3 - t)) - 80 (3 - t)

化简

3.4 (- 2.5 (3 - t) cos (0.4 (3 - t)) + 6.25 sin (0.4 (3 - t))) - 12.75 cos (0.4 (3 - t)) - 80 (3 - t) : 8.5 t cos (0.4 (- t + 3)) + 80 t - 38.25 cos (0.4 (- t + 3)) + 21.25 sin (0.4 (- t + 3)) - 240

= 8.5 t cos (0.4 (- t + 3)) + 80 t - 38.25 cos (0.4 (- t + 3)) + 21.25 sin (0.4 (- t + 3)) - 240

解答补常数

= 8.5 t cos (0.4 (- t + 3)) + 80 t - 38.25 cos (0.4 (- t + 3)) + 21.25 sin (0.4 (- t + 3)) - 240 + C

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