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1
1 Further Integration
1.26
Tangent substitutions or
tan
x
2
\tan\frac{x}{2}
substitutions
1.28
Reducing sine and cosine integrals
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1.27
Reducing
tan
x
2
\tan\frac{x}{2}
Secondly,
sin
x
=
2
sin
x
2
cos
x
2
=
2
tan
x
2
cos
2
x
2
\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}={2\tan\frac{x}{2}\cos^{2}\frac{x}{2}}
=
2
tan
x
2
sec
2
x
2
=
2
t
1
+
t
2
{=\frac{2\tan\frac{x}{2}}{\sec^{2}\frac{x}{2}}=\frac{2t}{1+t^{2}}}
and
cos
x
=
2
cos
2
x
2
-
1
=
2
sec
2
x
2
-
1
\cos x={2\cos^{2}\frac{x}{2}-1=\frac{2}{\sec^{2}\frac{x}{2}}-1}
=
2
1
+
t
2
-
1
=
2
-
1
-
t
2
1
+
t
2
=
1
-
t
2
1
+
t
2
.
{=\frac{2}{1+t^{2}}-1=\frac{2-1-t^{2}}{1+t^{2}}=\frac{1-t^{2}}{1+t^{2}}.}