![2 Aim - Evaluate I ut es +2 dt 83 ht Ia 7P et stat Ex 9 t 2+1 Interpoate I- w 2+1 EX 6 I - x3 1 [] [ as (x3)3 8 I Now Differe](http://img.homeworklib.com/questions/e4c491b0-2613-11ec-b25d-9774348b7c18.png?x-oss-process=image/resize,w_560)
9. (12 pts) Evaluate the following definite and indefinite integrals: (a) 12 dt d dx 1* 2 t Hint: Apply Part 1 of the Fundamental Theorem of Calculus and the Chain Rule. (b) /6 sin 20 de (sin2 @ + 2)2 -
A linear spring-mass system (without friction) satisfies m(d^2x/dt^2) = -kx, Derive that m/2 (dx/dt)^2 + k/2 x^2 = constant = E. Consider the initial value problem such that at t = 0, = x_0 and dx/dt = v_0. Evaluate E. Using the expression for conservation of energy, evaluate the maximum displacement of the mass from its equilibrium position. Compare this to the result obtained from the exact explicit solution.
Let S f(w)dt = 6, f(x)dx = -4, log(x)dt = 12, 9(x) dx = 9 Use these values to evaluate the given definite integral: -3 (f(x) f(x) + g(x)) dx
Evaluate the integral. dt 11 (1+) 5:2 (1+)? dt =D
13. Evaluate: (emcos x dx. Hint: Notice we see sin x and its derivative cosx. u=sin x is a good choice for substitution. 14. Evaluated as 15. Evaluate: x cos(x") sin(x)dx. Hint: Since the cosine function is taken to the 4n power, try u = cos(x).
Ince ft) dt, then Fx) = fx) over Find the derivative using the Fundamental Theorem of Calculus, part 1, which states that if (x) is continuous over an interval [a, b], and the function F(x) is defined by F(x) Tabl d dr dx
Calculate the first AND second derivative dy/dx and d^2y/dx^2
for the curve given by:
r(t) = t-t, y(t) = 3t - t
use the Laplace transform to solve the given system of differential equations dx dt dx dt dt dt x(0) 0, y(o)0 x(t) =
20. The graph shows h. Use the graph to evaluate: (a) [",(*)dx (b) [**(*)dx {(x)dx (d) "(x)dx
solve 1 and 2.
Evaluate the integral. 3T/4 1) rt/4 D) o B)-16 C) Find the derivative of the integral using the Second Fundamental Theorem of Calculus 2) y- cos nt dt D) cos (3)-1 C) sin (3) B) cos (x3) A) 6x5 cos (x3)
Evaluate the integral. 3T/4 1) rt/4 D) o B)-16 C) Find the derivative of the integral using the Second Fundamental Theorem of Calculus 2) y- cos nt dt D) cos (3)-1 C) sin (3) B)...