FA07Ma1aSol6%20(1)

# FA07Ma1aSol6%20(1) - Math 1a Fall Term 2007 SOLUTIONS TO...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 1a Fall Term 2007 SOLUTIONS TO HOMEWORK 6 Problem 1 (20 points) (Apostol, pg 217, # 26) (a) Show that Z π xf (sin x ) dx = π 2 Z π f (sin x ) dx. [Hint: u = π- x ] (b) Use part (a) to deduce the formula Z π x sin x 1 + cos 2 x dx = π Z 1 dx 1 + x 2 . Solution 1(a) Let I = Z π xf (sin x ) dx. Substitute u = π- x ; then x = π- u and dx =- du . As x increases from 0 to π , u decreases from π to 0, thus I = Z π ( π- u ) f (sin( π- u ))(- du ) =- Z π ( π- u ) f (sin( π- u )) du. Using sin( π- u ) = sin u and the fact that for an integrable function g we have R b a g ( x ) dx =- R a b g ( x ) dx , we get I = Z π ( π- u ) f (sin u ) du = Z π πf (sin u ) du- Z π uf (sin u ) du. Replacing u by x we see that the second integral is I again, thus I = π Z π f (sin x ) dx- I, that is I = π 2 Z π f (sin x ) dx. Solution 1(b) Let I = Z π x sin x 1 + cos 2 x dx. Putting f ( t ) = t/ (2- t 2 ), f (sin x ) = sin x 2- sin 2 x = sin x 1 + cos 2 x and the integral can be rewritten as I = Z π xf (sin x ) dx. Using part (a) we get I = π 2 Z π f (sin x ) dx. 1 2 SOLUTIONS TO HOMEWORK 6 Rewriting back the expression for f (sin x ) we get I = π 2 Z π sin x 1 + cos 2 x dx. Substitute u =- cos x ; then du = sin x dx . Further, as x increases from 0 to π we see that u increases from- cos(0) =- 1 to- cos( π ) = 1. So I = π 2 Z 1- 1 du 1 + u 2 . Notice that g ( u ) = du 1+ u 2 is an even function of u , in other words g (- u ) = g ( u ). Since for even functions R a- a g ( u ) du = 2 R a g ( u ) du (split into two pieces, substitute u by- t in one of them, change back t by u ), we get I = ( π 2 )2 Z 1 du 1 + u 2 = π Z 1 du 1 + u 2 . Replacing the variable u by the variable x , I = π Z 1 dx 1 + x 2 , which is what we set out to deduce....
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

FA07Ma1aSol6%20(1) - Math 1a Fall Term 2007 SOLUTIONS TO...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online