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Mathematics
Mathematics
Problem
Consider the integral involving tangent and secant. (i) Evaluate the indefinite integral
∫
tan
(
a
x
+
b
)
sec
2
(
a
x
+
b
)
d
x
\displaystyle \int \tan(ax+b)\,\sec^{2}(ax+b)\,dx
∫
tan
(
a
x
+
b
)
sec
2
(
a
x
+
b
)
d
x
for
a
≠
0
a\ne 0
a
=
0
. (ii) Hence, compute
K
=
∫
0
π
/
4
tan
(
2
x
+
π
6
)
sec
2
(
2
x
+
π
6
)
d
x
K=\displaystyle \int_{0}^{\pi/4}\tan\bigl(2x+\tfrac{\pi}{6}\bigr)\,\sec^{2}\!\bigl(2x+\tfrac{\pi}{6}\bigr)\,dx
K
=
∫
0
π
/4
tan
(
2
x
+
6
π
)
sec
2
(
2
x
+
6
π
)
d
x
. (iii) Express your answer for
K
K
K
as a rational multiple of
tan
2
(
π
3
)
\tan^{2}\!(\tfrac{\pi}{3})
tan
2
(
3
π
)
.
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