365
Answers
501–600
Answers and Explanations
Therefore, the integral becomes
x
dx
x
dx
x
dx
x dx
x
dx
−
= − −
+
−
=
−
+
−
=
−
−
−
∫
∫
∫
∫
∫
2
2
2
2
2
2
3
4
2
4
3
2
3
2
2
4
(
)
(
)
(
)
(
)
x x x
x
x
−
+
−
=
−
−
− −
−
−
2
3
2
2
2
4
2
2
2
2
2
2 2 2
2
2 3
3
2
( )
( )
( )
+
−
−
−
= − − −
4
2
2 4
2
2
2 2
4 2
6
2
2
( )
( )
(
)
− −
( ) + − − −
=
9
2
8 8
2 4
29
2
(
) (
)
571.
4
7
7 4
x
C
+
Add 1 to the exponent and divide by that new exponent to get the antiderivative:
x dx x
C
x
C
3 4
7 4
7 4
7 4
4
7
∫
=
+
=
+
572.
−
+
5
6
6
x
C
First move the variable to the numerator:
5
5
5
7
7
7
x
dx
x dx
x dx
∫
∫
∫
=
=
−
−
Then integrate by adding 1 to the exponent while dividing by that new exponent:
5
5
6
5
6
7
6
6
x dx
x
C
x
C
−
−
∫
=
−
+
= −
+
573.
x
x
x C
4
3
4
+
− +
Find the antiderivative of each term:
x
x
dx x
x
x C
x
x
x C
3
2
4
3
4
3
3
1
4
3 3
4
+
−
(
) = + − +
=
+
− +
∫
574.
3 sin x + 4 cos x + C
Use elementary antiderivative formulas:
( cos
sin )
sin
cos
3
4
3
4
x
x dx
x
x C
−
=
+
+
∫
Answers
501–600
Answers and Explanations
Therefore, the integral becomes
x
dx
x
dx
x
dx
x dx
x
dx
−
= − −
+
−
=
−
+
−
=
−
−
−
∫
∫
∫
∫
∫
2
2
2
2
2
2
3
4
2
4
3
2
3
2
2
4
(
)
(
)
(
)
(
)
x x x
x
x
−
+
−
=
−
−
− −
−
−
2
3
2
2
2
4
2
2
2
2
2
2 2 2
2
2 3
3
2
( )
( )
( )
+
−
−
−
= − − −
4
2
2 4
2
2
2 2
4 2
6
2
2
( )
( )
(
)
− −
( ) + − − −
=
9
2
8 8
2 4
29
2
(
) (
)
571.
4
7
7 4
x
C
+
Add 1 to the exponent and divide by that new exponent to get the antiderivative:
x dx x
C
x
C
3 4
7 4
7 4
7 4
4
7
∫
=
+
=
+
572.
−
+
5
6
6
x
C
First move the variable to the numerator:
5
5
5
7
7
7
x
dx
x dx
x dx
∫
∫
∫
=
=
−
−
Then integrate by adding 1 to the exponent while dividing by that new exponent:
5
5
6
5
6
7
6
6
x dx
x
C
x
C
−
−
∫
=
−
+
= −
+
573.
x
x
x C
4
3
4
+
− +
Find the antiderivative of each term:
x
x
dx x
x
x C
x
x
x C
3
2
4
3
4
3
3
1
4
3 3
4
+
−
(
) = + − +
=
+
− +
∫
574.
3 sin x + 4 cos x + C
Use elementary antiderivative formulas:
( cos
sin )
sin
cos
3
4
3
4
x
x dx
x
x C
−
=
+
+
∫