BC 2,3
Problem Set #1
(Due Friday, August 30)
Name: ______________
Please show appropriate work – no calculator or computers – except to check work done by hand.
Work should be shown clearly, using correct mathematical notation. Please show enough work on all
problems so that others could follow your work and do a similar problem without help. Collaboration
is encouraged, but in the end, the work should be your own.
1.
Let a  0 be a real constant. Let R be the region in the first quadrant bounded by

y  tan 1 x, y  , x  a , and the y–axis.
2
a. Sketch the region R. [Label scales on both axes. Neatness and accuracy count].
b. Find the exact area of R in terms of a.
c. If A(a ) represents the area in part b. as a function of a, evaluate lim  A(a) .
a 
BC 2,3
Problem Set #1
(Due Friday, August 30)
Name: ______________
x4
 5
2. Find the minimum distance from the point  0,  to the graph of y  .
8
 2
3. Evaluate:
a.
2
 x 1  x dx
b.
sin 2 x
 cos4 x dx
BC 2,3
Problem Set #1
(Due Friday, August 30)
Name: ______________
I. Look at integrals involving trigonometric powers of the form  sin n x cosm x dx where n.m .
i. If m is odd, let u  sin x . Then du  cos x dx and we can rewrite the integral as follows:
n
m
n
m 1
 sin x cos x dx   sin x cos x cos x dx .
m1
m1
Since m – 1 is even, cosm1 x  (cos2 x) 2  (1  sin 2 x) 2 . Thus,
n
m
n
m 1
 sin x cos x dx   sin x cos x cos x dx
  sin n x(1  sin 2 x)
  u n (1  u 2 )
Since m – 1 is even, u n (1  u 2 )
m1
2
m 1
2
m 1
2
cos x dx
du
is merely a polynomial and the integration is simple.
ii. If n is odd, let u  cos x . Then du   sin x dx and we can rewrite the integral as follows:
n
m
n 1
m
 sin x cos x dx   sin x cos x sin x dx .
Using a technique similar to i. , we can integrate.
iii. What happens if both n,m are even? We use the identities
1  cos 2 x
1  cos 2 x
sin 2 x 
and cos 2 x 
2
2
to reduce the powers of sine and cosine. This gets a bit messy when n,m are large.
Example:
 sin x cos x dx 
4
2





 1  cos 2 x   1  cos 2 x 

 
 dx
2
2

 

1

4
1
 
4

2
 cos
3
2 x  cos 2 2 x  cos 2 x  1 dx
1  cos 2 x
 2

 cos 2 x  1  dx
 cos 2 x  cos 2 x 
2


11
1
1
1

  cos3 2 x  x  sin 2 x  sin 2 x  x  C 
4 6
2
4
2

BC 2,3
Problem Set #1
(Due Friday, August 30)
Name: ______________
II. Using the above explanation as a model, explain how to integrate  tan n x secm x dx where n.m .
i. If m is even.
ii. If n is odd