# How do you find the definite integral of #tan x dx# from #[pi/4, pi]#?

Actual Value: Impossible to determine

Value by Cauchy's Principal Value:

However, if we use Cauchy principal value, we can derive the above answer.

This is currently very hard to integrate. Let's attempt a u-substitution.

Hopefully this helps!

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To find the definite integral of tan(x) dx from π/4 to π, you can use the properties of the tangent function and integration techniques. Firstly, rewrite tan(x) in terms of sine and cosine using the identity tan(x) = sin(x)/cos(x). Then, apply the substitution method, letting u = cos(x) and du = -sin(x) dx. This transforms the integral into ∫(1/u) du. Integrate this expression with respect to u, which yields ln|u|. Now, substitute back u = cos(x), resulting in ln|cos(x)|. Finally, evaluate this expression from π/4 to π and subtract the lower limit value from the upper limit value. This process will give you the definite integral of tan(x) dx from π/4 to π.

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