Part II: The Answers
374
Answers
601–700
x
x
x
dx
x x
x
dx
x x
x
x
dx
x x
dx
3
2
25
5
25
5
5
5
5
5
−
+
=
−
(
)
+
=
+
−
+
=
−
∫
∫
∫
(
)
(
)(
)
(
)
(
)
∫ ∫
∫
=
−
(
)
=
−
+
x
x dx
x
x C
2
3
2
5
3
5
2
607.
tan x + C
Begin by writing tan
2 
x as sin
cos
2
2
x
x
. Then simplify:
tan
sin
sin
cos
sin
cos
sec
2
2
2
2
2
2
2
1
1
x
x
dx
x
x
x
dx
x
dx
x dx
∫
∫
∫
∫
=






=
=
= =
+
tan x C
608.
sin x + cos x + C
Use a trigonometric identity on the numerator of the integrand, factor,
and simplify:
cos
cos
sin
cos
sin
cos
sin
(cos
sin )(cos
2
2
2
x
x
x
dx
x
x
x
x
dx
x
x
x
+
=
−
+
=
−
∫
∫
+ +
+
=
−
=
+
+
∫
∫
sin )
cos
sin
(cos
sin )
sin
cos
x
x
x
dx
x
x dx
x
x C
609.
–cot x – 2x + C
Begin by using a trigonometric identity on the numerator of the integrand. Then
split up the fraction and rewrite both fractions using trigonometric identities:
cos
sin
cos
sin
cos
sin
sin
2
2
1
2
1
2
2
2
2
2
2
x
x
dx
x
x
dx
x
x
x
dx
∫
∫
∫
=
−
=
−






= =
−
(
)
∫ 2
2
2
cot
csc
x
x dx
Notice that the first term of the integrand does not have an elementary
antiderivative, so you can use a trigonometric identity again to simplify
and then integrate: