Calculus: Analytic Geometry and Calculus, with Vectors

522 Exponential and logarithmic functions

3 Derive the formula

fx" cos x dx = x" sin x + X"-1 cos x - n(n - I) fx"-2 cos x dx.

Use this formula to find fx2 cos x dx, and check the result by differentiation.

This formula and those of the next two problems are reduction formulas. In

some cases, useful results are obtained by making repeated application of them.

4 Letting

I(p,q) = f sin' x cosq x dx,

where p and q are constants for which p -1 and p + q P- 0, show that

I(p

q)

sin'+1 x cost"'' x+ q = 1

s i n x cosq-2 x dx

p+1 p+1J

and

sinp+, x cosq-1 x q - 1

I(p,q) = p+ q p+

q

f sinp xcosq-2 x dx.

Hint: Start by writing u = cosq-' x, dv = sin' x cos x dx.

5 Letting

W(p,q) = f sin' x cosq x dx,

where p and q are constants for which q , -1 and p + q ; 0, show that

W(p,q) = -

and

sin-q+^11 q+1x+q+1 11

f sinp-2x cos9+2xdx

sin-1 x cosq+' x p

W(p,q) p + q +p + q f sin'-' x cosq x dx.

Hint: Start by writing u = sin'-' x, dv = cosq x sin x dx.

6 Supposing that p is an integer for which p 2, show how a result of the

preceding problem can be used to obtain the formula

(1) I

_ 1

x

o

sin'-' x d.,

Supposing that n is a positive integer, show how repeated applications of (1)

give the famous Wallis (1616-1703) formulas

(2) '/' 2n-1 2n-3 5 3 1 wr

Io sin2" x dx = 2n 2n - 2 - 6 4 2 2

;/2 2n 2n-2

v

(3) sin2n+' x dx^642
0 =2n+l 2n-1
7 5 3
Observe that

(4) 2n(2n - 2). .6-4-2 = 2"n!