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Q:

$\int_0^1 \frac{\sec^2x-\tan^2x}{\tan x}dx=-\ln\sin\left(\frac{\pi}{8}\right)$

Evaluate the following integral.
$$\int_0^1 \frac{\sec^2x-\tan^2x}{\tan x}dx=-\ln\sin\left(\frac{\pi}{8}\right)$$
I have checked with Mathematica, but I don’t know how to evaluate this integral. Can you guys help?

A:

This is the integral representation of arctan.
$$
\arctan(x)=\int_0^x\frac{\mathrm dt}{1+t^2}
$$
Make the substitution $t=\tan x$, then $\mathrm dt=\sec^2x \,\mathrm dx$, and rewrite the integral:
$$
\int_0^1 \frac{\sec^2x-\tan^2x}{\sec x} \,\mathrm dx=-\int_0^1\frac{\mathrm dt}{\sec t}
$$
Consider your integral, then recall the integral representation of arctan again.
$$
\frac{1}{1+\tan^2x}=\int_

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