373
Answers
601–700
Answers and Explanations
603.
x + C
Start by using a trigonometric identity for 1 + cot
2 
x, followed by rewriting csc
2 
x as
1
2
sin x
. Then simplify:
sin
c ot
sin
csc
sin
sin
2
2
2
2
2
2
1
1
1
x
x dx
x
x dx
x
x
dx
dx
+
(
) =
(
)
=






=
∫
∫
∫
∫ ∫
= +
x C
604.
4
5 1
2
x x x
C
− +
+
Begin by splitting the fraction into three terms:
4
5
2
4
5
2
4 5
2
3
3
3
3
3
3
2
3
x
x
x
dx
x
x
x
x
x
dx
x
x d
+
−





 =
+
−






=
+
−
(
)
∫
∫
−
−
x x
∫
Then use elementary antiderivative formulas and simplify:
4 5
2
4
5
1
2 2
4
5 1
2
3
1
2
2
+
−
(
) = + −
− −
+
=
− +
+
−
−
−
−
∫
x
x dx
x
x
x
C
x x x
C
605.
7x + C
Begin by factoring the 3 out of the first two terms of the integrand. Then use a trigonometric identity:
3
3
4
3
4
3 1 4
7
2
2
2
2
sin
cos
sin
cos
( )
x
x
dx
x
x
dx
dx
dx
+
+
(
)
=
+
(
) +
(
)
=
+
(
)
=
∫
∫
∫
= =
+
∫
7x C
606.
x
x C
3
2
3
5
2
−
+
Begin by factoring the numerator. Then simplify and use elementary antiderivative
formulas: