MATH 1101Y 2009 Quiz 10 (a) Evaluate the indefinite integrals. 1

MATH 1101Y 2009 Quiz 10 (a) Evaluate the indefinite integrals. 1

MATH 1101Y 2009 Quiz 10 (a)
Evaluate the inde…nite integrals.
R
8
1. (1.5 pts) x3 (x4 + 6) dx
Solution: Let u = x4 + 6; du = 4x3 dx:
Z
8
x3 x4 + 6 dx
Z
1
8
=
4x3 x4 + 6 dx
4
Z
1
1 u9
=
u8 du =
+C
4
4 9
9
(x4 + 6)
+ C:
=
36
2. (1.5 pts)
R
ln x
dx
x
Solution: Let u = ln x: du = x1 dx:
Z
=
Z
ln x
dx
x
udu =
u2
+C
2
(ln x)2
+ C:
=
2
3. (2 pts)
R
x
dx
1+x4
Solution: Let u = x2 : du = 2xdx:
Z
=
=
=
=
x
dx
1 + x4
Z
1
2x
dx
2
1 + x4
Z
1
1
du
2
1 + u2
1
arctan (u) + C
2
1
arctan x2 + C:
2
1