Answer:
-30√(t/a) cos(√(at)) + 30/a sin(√(at)) + C
Step-by-step explanation:
∫ 15 sin(√(at)) dt
Use substitution:
If x = √(at), then:
dx = ½ (at)^-½ (a dt)
dx = a / (2√(at)) dt
dx = a/(2x) dt
dt = (2/a) x dx
Plugging in:
∫ 15 sin x (2/a) x dx
30/a ∫ x sin x dx
Integrate by parts:
If u = x, then du = dx.
If dv = sin x dx, then v = -cos x.
∫ u dv = uv − ∫ v du
= 30/a (-x cos x − ∫ -cos x dx)
= 30/a (-x cos x + ∫ cos x dx)
= 30/a (-x cos x + sin x + C)
Substitute back:
30/a (-√(at) cos(√(at)) + sin(√(at)) + C)
-30√(t/a) cos(√(at)) + 30/a sin(√(at)) + C
Answer:
Step-by-step explanation:
Factor constant terms:
Divide both sides by -2:
Expand trigonometric functions using the fact:
So:
Factor sin(x) and constant terms and multiply both sides by -1:
Split into two equations:
For (1)
Add 1 to both sides and divide both sides by 8:
Take the inverse cosine of both sides:
For (2)
Simply take the inverse sine of both sides
Therefore, the solutions are given by:
Answer:
b
Step-by-step explanation:
4800-1400 because 200*7 equals 1400.