Part II: The Answers
366
Answers
501–600
575.
4x + C
This problem can be a bit difficult, unless you remember the trigonometric identities!
In this problem, simply factor out the 4 and use a trigonometric identity:
4
4
4
4 1
4
4
2
2
2
2
cos
sin
cos
sin
( )
x
x dx
x
x dx
dx
dx
x C
+
(
) =
+
(
)
=
=
=
+
∫
∫
∫
∫
576.
tan x – x + C
Use a trigonometric identity followed by elementary antiderivatives:
tan
s ec
tan
2
2
1
x dx
x
dx
x x C
∫
∫
=
−
(
)
=
− +
577.
x
3
+ x
2
+ x + C
Find the antiderivative of each term:
3
2 1
3 3
2 2
2
3
2
3
2
x
x
dx
x
x
x C
x
x
x C
+
+
(
) = + + +
=
+ + +
∫
578.
2
7
7 3
5
x
x C
+
+
Use elementary antiderivative formulas:
2
3
5
2
3 7 3
5 5
2
7
4 3
4
7 3
5
7 3
5
x
x dx
x
x
C
x
x C
+
(
) =








+





 +
=
+ +
∫
579.
4
2
2
x x
x C
+
+
+
tan
Begin by using a trigonometric identity to replace tan
2 
x. Then simplify:
5
5
1
4
4
2
2
2
2
2
+ +
(
) = + +
−
(
)
=
+ +
(
)
=
+
+
∫
∫
∫
x
x dx
x
x
dx
x
x dx
x x
x
tan
s ec
sec
tan + + C