Part II: The Answers
368
Answers
501–600
583.
1
3
x C
+
Use a trigonometric identity in the numerator of the integrand and simplify:
1
3
1
3
1
3
1
1
3
2
2
2
2
−
=
=
=
+
∫
∫
∫
sin
cos
cos
cos
x
x
dx
x
x
dx
dx
x C
584.
x
x
C
5
4
5
3
4
+
+
Begin by multiplying out the integrand. Then integrate each term:
x x
x dx
x
x dx
x
x
C
x
x
C
2
2
4
3
5
4
5
4
3
3
5
3 4
5
3
4
+
(
) =
+
(
)
=
+
+
=
+
+
∫
∫
585.
− −
−
+
1
1
2
1
3
2
3
x
x
x
C
Begin by breaking up the fraction and simplifying each term:
x
x
x
dx
x
x
x
x
x
dx
x
x
x dx
2
4
2
4
4
4
2
3
4
1
1
+ +
=
+
+

 

 
=
+
+
(
)
∫
∫
∫
−
−
−
Then find the antiderivative of each term:
x
x
x dx x
x
x
C
x
x
x
C
−
−
−
−
−
−
+
+
(
) = −
+ −
+ −
+
= − −
−
+
∫
2
3
4
1
2
3
2
3
1
2
3
1
1
2
1
3
586.
x
x
x C
+
+
+
2
3
1
5
3
5
Expand the integrand and use elementary antiderivative formulas:
1
1
1
1 2
2
3
1
5
2
2
2
2
2
4
3
5
+
(
) = +
(
) +
(
)
=
+
+
(
)
= +
+
+
∫
∫
∫
x
dx
x
x dx
x
x dx
x
x
x C
587.
3x + cot x + C
Use a trigonometric identity on the integrand and simplify: