A particle is moving with the given data. Find position of the particles

Question

A particle is moving with the given data. Find position of the particles

a(t) = 13sin(t)+7cos(t), s(0) = 0, s(2pi) = 10

i got

s(t) = -13sin(t) -7cos(t)+5pi+7 but it was wrong

thabk you for help

Answer 1

function a is the acceleration

s is the speed

s(t) = integral a(t)

= integral [ 13 sin(t) - 7 cos(t) ]

= -13 cos(t) - 7 sin(t) + C

s(0) = 0

0 = -13 + C

C = 13

s(t) = -13 cos(t) - 7 sin(t) + 13

x(t) = integral s(t)

= integral -13 cos (t) - 7 sin(t) + 13

= -13 integral cos(t) - 7 integral sin(t) + 13 t + C

-13 sin (t) + 7 cos(t) + 13t + C

Assuming the initial position at zero:

x(0) = 0 ----> 0 = 7 + C

C = -7

x(t) = -13 sin t + 7 cos t + 13 t - 7