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1 Further Integration
1.28
Reducing sine and cosine integrals
1.30
Continuing Wright’s integral (1599)
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1.29
Wright’s integral (1599)
Example
.
∫
sec
x
d
x
=
log
|
1
+
tan
x
2
1
-
tan
x
2
|
+
C
.
\int\sec x\,dx=\log\left|{{1+\tan\frac{x}{2}}\over{1-\tan\frac{x}{2}}}\right|+C.
Solution.
We make the above subsitutions
∫
sec
x
d
x
=
∫
1
cos
x
d
x
=
∫
1
+
t
2
1
-
t
2
2
d
t
1
+
t
2
\int\sec x\,dx={\int{{1}\over{\cos x}}{{dx}}}\,{=\int{{1+t^{2}}\over{1-t^{2}}}% {{2dt}\over{1+t^{2}}}}
=
∫
2
d
t
1
-
t
2
=
∫
(
1
1
-
t
+
1
1
+
t
)
d
t
{=\int{{2dt}\over{1-t^{2}}}}\,{=\int\Bigl({{1}\over{1-t}}+{{1}\over{1+t}}\Bigr% )dt}
=
-
log
|
1
-
t
|
+
log
|
1
+
t
|
+
C
=
-
log
(
1
-
tan
x
2
)
+
log
(
1
+
tan
x
2
)
+
C
.
{=-\log|1-t|+\log|1+t|+C}\,{=-\log(1-\tan\frac{x}{2})+\log(1+\tan\frac{x}{2})+% C.}