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.

1. 3

2. 4

3. 1

hope this ! have a great day.

-wfz

none.

step-by-step explanation:

m

1

=

5

m1=5

b

=

−

1

b=-1

find the slope and y-intercept of the second equation.

tap for more

m

2

=

0

m2=0

b

=

−

6

b=-6

compare the slopes

m

m of the two equations.

m

1

=

5

,

m

2

=

0

m1=5,m2=0

compare the two slopes in the decimal form. if the slopes are equal, then the lines are parallel. if the slopes are not equal, then the lines are not parallel.

m

1

=

5

,

m

2

=

0

m1=5,m2=0