369
Answers
501–600
Answers and Explanations
2
2
1
3
3
2
2
2
−
(
) = −
−
(
)
(
)
=
−
(
)
=
+
+
∫
∫
∫
cot
c sc
csc
cot
x dx
x
dx
x dx
x
x C
588.
2
2
2
x
x C
−
+
Begin by splitting up the fraction and writing the radical using exponential notation:
x
x
x
dx
x
x
x
x
dx
x
x dx
−
=
−






=
−
(
)
∫
∫
∫
−
4
4
4
2
1 2
2
1 2
Then use elementary antiderivative formulas and simplify:
x
x dx x
x
C
x
x C
−
−
(
) = − +
=
−
+
∫
1 2
1 2
2
2
4
1 2
4 2
2
2
589.
3
5
3
2
5 3
4 3
x
x
x C
+
+ +
Begin by writing the cube root using exponential notation and multiplying out the
integrand:
x
dx
x
x
d x
x
x
dx
3
2
1 3
1 3
2 3
1 3
1
1
1
2
1
+
(
) =
+
(
) +
(
)
=
+
+
(
)
∫
∫
∫
Then use elementary antiderivative formulas:
x
x
dx x
x
x C
x
x
x C
2 3
1 3
5 3
4 3
5 3
4 3
2
1
5 3
2 4 3
3
5
3
2
+
+
(
) = +
+ +
=
+
+ +
∫
590.
2
3
6
7
3 2
7 6
x
x
C
+
+
Begin by rewriting the radicals using exponential notation and then distribute:
x
x
dx
x
x
dx
x
x
dx
x
x
∫
∫
∫
+






=
+






=
+
(
)
=
+
(
−
1 1
1
1
1
3
1 2
1 3
1 2
1 3
1 2
1 6
) )
∫
dx
Next, use elementary antiderivative formulas for each term:
x
x
dx x
x
C
x
x
C
1 2
1 6
3 2
7 6
3 2
7 6
3 2
7 6
2
3
6
7
+
(
) = + +
=
+
+
∫