# Exercises5Sol

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```					Exercise #5 for Calc II.

1.    x2 sin x dx

x2 sin x dx   =      x2 d(− cos x)

= x2 (− cos x) −          (− cos x) dx2

= x2 (− cos x) +           2x cos x dx

= −x2 cos x +            2x d sin x

= −x2 cos x + 2x sin x −               sin x d2x

= −x2 cos x + 2x sin x − 2              sin x dx
= −x2 cos x + 2x sin x + 2 cos x + C

2.    sin x · ex dx.

sin x · ex dx   =      sin x dex

=   sin x · ex −       ex d sin x

=   sin x · ex −       ex · cos x dx

=   sin x · ex −       cos x dex

=   sin x · ex − (cos x · ex −         ex d cos x)

=   sin x · ex − cos x · ex −         ex · sin x dx

sin x · ex − cos x · ex
Solve for    sin x · ex dx, we get    sin x · ex dx =                           + C.
2

3.    sin3 x dx

sin3 x dx =            sin2 x · sin x dx

=           sin2 x d(− cos x)

= −           1 − cos2 x d cos x
= ···

1
x+1
4.                         dx.
x2   + 2x + 2
x+1                                  x+1
dx =                                    dx
x2   + 2x + 2                      (x2  + 2x + 1) + 1
(x + 1)
=                           d(x + 1)
(x + 1)2 + 1
=        ···

x
5.       tan         dx.
2
x          sin x
2
intxtan       =             dx
2         cos x2
sin u               x
=            d2u (set u = )
cos u               2
sin u
= 2            du
cos u
1
= 2            d(− cos u)
cos u
= ···

2
6.                2x − x2 dx. (Hint: Trigonometric substitution.)
0
√
Note that              2x − x2 =   1 − (1 − 2x + x2 ) =     1 − (x − 1)2 . Let x − 1 = sin t, we get · · ·

√
7. Let D be the area formed by y = 2x − x2 (0 ≤ x ≤ 2) and x-axis. If we rotate D around y-axis,
what is the volume of solid we will get? (A bit challenging, but doable)
We can use cylindrical cell to ﬁgure out the volume. In fact,
2
Volume   =              2πx 2x − x2 dx ,
0

then you can apply u-substitution to get                 2πx 2x − x2 dx.

8. The force needed to pull or compress a spring is given by F = kd, with k being the Hook’s constant
and d being the oﬀset of the spring from the natural length. In the beginning, the spring is at natural length
L0 . What is the work needed to pull the spring to a length of L (with L > L0 )?

2
L
Work    =            F dx
L0
L
=            k(x − L0 ) dx
L0
1
=     k(x − L0 )2        L
L0
2
1
=     k(L − L0 )2
2

x
9. How to ﬁnd 1 1 dt? Is this value related to x? When x is approaching ∞, what can you say about
t
x 1
1 t
dt?
∞                              ∞                                       ∞
Generally speaking, by a f (t) dt, we mean limx→∞ a f (t) dt. With that in mind, what is 1 1 dt?      t
∞                 ∞
What is 1 t1 dt? What is 0 e−x dt?
2
1                              1
If f is continuous every where except at 0, for example, f (x) = x . How would you try to deﬁne 0 f (t) dt?
1               1 1
For example, what is 0 1 dt? What is 0 √t dt?
t

Hint: This problem is covered in class. If you have questions on this problem, drop by my oﬃce on
Monday.

10. Find the area bounded by y = x and y = xn (with 0 ≤ x ≤ 1, and n is a natural number). When
the value of n increases, will the area increase or decrease? Sketch the graph for n = 2 and n = 3, explain
(graphically) why the area increases if we increase the value of n.

1
Area   =            x − xn dx
0
x2    xn+1 1
= ( −          )
2    n+1 0
1      1
=     −
2 n+1
When n increases, the area will decrease. You can easily see this on the graph.

3

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