How to integrate
How to integrate sin6 x cos2 x dx
I do not know of an easy way to evaluate this integral. But here's one way that isn't too bad. It repeatedly uses the reduction formula
sinn x dx = - 1 cos x sinn-1 x + n - 1 sinn-2 x dx
n
n
where n 2 is an integer,
which is proved as Example 6 in Section 6.1.
sin6 x cos2 x dx = sin6 x(1 - sin2 x) dx
[ using sin2 x + cos2 x = 1 ]
= sin6 x dx - sin8 x dx
=
sin6 x dx -
-
1 8
cos
x
sin7
x
+
7 8
sin6 x dx
[ reduction formula ]
=
sin6
x
dx
+
1 8
cos
x
sin7
x
-
7 8
sin6 x dx
=
1 8
cos x sin7
x
+
1 8
sin6 x dx
[ combining like integrals ]
=
1 8
cos x sin7
x
+
1 8
-
1 6
cos
x
sin5
x
+
5 6
sin4 x dx
[ reduction formula ]
=
1 8
cos x sin7
x
-
1 48
cos x sin5
x
+
5 48
sin4 x dx
=
1 8
cos x sin7
x
-
1 48
cos x sin5
x
+
5 48
-
1 4
cos
x
sin3
x
+
3 4
sin2 x dx
[ reduction formula ]
=
1 8
cos x sin7
x
-
1 48
cos x sin5
x
-
5 192
cos x sin3
x
+
5 64
sin2 x dx
=
1 8
cos x sin7
x
-
1 48
cos x sin5
x
-
5 192
cos x sin3
x
+
5 64
?
1 2
(1 - cos 2x) dx
[ using
sin2
x
=
1 2
(1
-
cos 2x) ]
=
1 8
cos x sin7
x
-
1 48
cos x sin5
x
-
5 192
cos x sin3
x
+
5 128
x
-
1 2
sin
2x
+C
[ mental substitution u = 2x ]
=
1 8
cos x sin7
x
-
1 48
cos x sin5
x
-
5 192
cos x sin3
x
+
5 128
x
-
5 256
sin 2x
+
C.
Compare Example 3 in Section 6.2, which shows how to integrate sin2 x (the same method is used above).
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