Use the substitution x = 2 − cos θ to evaluate the integral ∫ 2 3/2 ( x − 1 3 − x )1 2 dx. Show that, for a < b, ∫ q p ( x − a b − x )1 2 dx = (b − a)(π + 3√ 3 − 6) 12 , where p =

If the integral as written in my comment is accurate, then we have

Expand the polynomial, then complete the square within the square root:

Let and :

Recall that for all , but for all in the integration interval we have . So :

Recall the double angle identity,

You can determine the more general result in the same way.

Complete the square to get

and let for brevity. Note that

Make the following substitution,

and the integral reduces like before to

where

(Depending on the interval [p, q] and thus [P, Q], the square root of cosine squared may not always reduce to sine.)

Resolving the integral and replacing c, with

because , gives

Without knowing p and q explicitly, there's not much more to say.

whats the question?

step-by-step explanation:

answer:   35 % is the probability of a late departure or an early arrival.

step-by-step explanation:

let l represents the late departure and e represents the early departure.

then, according to the question,

p(l) = 12 %,   p(e) = 27 % and = 4 %

and, we have to find out,

but, we know that,

which is the required equation that will use to find the probability of a late departure or an early departure.

therefore,   the probability of a late departure or an early arrival =   35 %