Practice 3.1.1: Indefinite Integrals
1. Evaluate the following indefinite integrals:
i.
x xdx
∫
ii.
( )
2
2x
dx
x
−
∫
iii.
cos
1 2sin
x
dx
x
−
∫
SOLUTION:
i.
5
2
5
x
C
+
ii.
( )
2
23 20 60
15
x
x x C
− + +
iii.
(
)
0.5ln 1 2sin
x C
− − +
2.
Find the primitives of the following functions passing through at points indicated:
i.
( )
1
1
f t t
t
= +
− at point
(
)
2,0
ii.
( )
2
2 3
x
f x
x
=
+
at point
ln3
0,
4
SOLUTION:
i.
( ) ( )
2
ln 1 2
2
t
f t t
= − + −
ii.
( )
( )
2
1
ln 2 3
4
f x x
= +