Use the substitution x=3 cos t, 0 ≤ t ≤ π to simplify the following integral:

Question

Use the substitution  x=3 cos t, 0 ≤ t ≤ π  to simplify the following integral:

∫ 1 / sqrt(9 - x^2)

(a) Calculate sqrt(9 - x^2) in terms of t.

(b) If the substitution replaces dx with f(t) dt then what is the function f(t)?

(c) Hence write the integral in terms of t: ∫ dt

(d) Perform this integral, including constant of integration c.

(e) Convert your answer from a function of t to a function of x.

Answer 1

x = 3cost So, dx = -3sintdt

∫ [1 / √(9 - x²)]dx = -∫ [1 / √(9 - 9cos²t)](3sint)dt

= -∫ [1 / √(1 - cos²t)]sintdt = -∫[1 / √sin²t]sintdt (When 0 < t < π, sint > 0, so √sin²t = l sint l = sint)

= -∫ dt = -t + C Since x = 3cost, cost = x/3. So, t = Cos^-1(x/3).

= -Cos^-1(x/3) + C