MATH 1101Y 2009 Quiz 13 (a) Evaluate the integral. 1. (2 pts) ∫ x

MATH 1101Y 2009 Quiz 13 (a)
Evaluate the integral.
R p
1. (2 pts) x 9 x2 dx
Solution 1 : Let x = 3 sin : dx = 3 cos d :
p
p
p
9 x2 =
9 9 sin2 = 3 1
p
= 3 cos2 = 3 cos :
Z
p
x 9
x2 dx
=
sin2
Z
3 sin 3 cos 3 cos d
Z
= 27 sin cos2 d (Let u = cos : du = sin d :)
Z
u3
=
27 u2 du = 27 + C = 9 cos3 + C
3
=
9
(9
3
x2 ) 2
33
+C =
1
9
3
x2
3
2
+ C:
3
x
θ
9 − x2
Solution 2 : Let u = 9 x2 : du = 2xdx:
Z
Z p
p
1
2
( 2x) 9 x2 dx
x 9 x dx =
2
Z
1 p
12 3
=
u2 + C
udu =
2
23
3
1
=
9 x2 2 + C:
3
2. (3 pts)
R
3
px
dx
1+x2
Solution 1 : Let x = tan : dx = sec2 d :
p
p
1 + x2 =
1 + tan2
p
=
sec2 = sec :
1
Z
Z
x3
tan3 sec2
p
d
dx =
sec
1 + x2
Z
=
tan3 sec d Let u = sec : du = tan sec :
Z
Z
2
=
tan tan sec d =
sec2
1 tan sec d
Z
u3
=
u2 1 du =
u+C
3
3
sec3
(1 + x2 ) 2 p
=
sec + C =
1 + x2 + C:
3
3
Solution 2 : Let u = 1 + x2 : du = 2xdx: x2 = u 1:
Z
Z 2
Z
x3
x (2x) dx
u 1
1
1
p
p
p du
dx =
=
2
2
u
1 + x2
1 + x2
Z
p
1
1 2 3
1
u p
du =
u2
=
2
2 3
u
p
3
1
1 + x2 2
1 + x2 + C:
=
3
2
1
2u 2
+C