Question

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

Answer 1

Actual Value: Impossible to determine

Value by Cauchy's Principal Value: `int_(pi/4)^pitanxdx = ln(1/sqrt(2)) = -0.34657`

Explanation:

For a simple answer, this integral doesn't converge (it has an asymptote at `x = pi/2`.

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

We are asked to find `int_(pi/4)^pi tanx dx`. The first step in solving this problem is determining the antiderivative.

Recall that `tanx = sinx/cosx`:

`= int_(pi/4)^pi sinx/cosx dx`

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

Let `u = cosx`. Then `du = -sinxdx` and `dx = -(du)/sinx`

`=int_(pi/4)^pi sinx/u * -(du)/sinx`

`=int_(pi/4)^pi -1/udu`

By the rule `int1/xdx = ln|x| + C`.

`=-[ln|u|]_(pi/4)^pi`

`= -[ln|cosx|]_(pi/4)^pi`

Evaluate using the second fundamental theorem of calculus, which states that `int_a^bF(x)dx = f(b) - f(a)`, where `int(F(x))dx = f(x)`.

`=-(ln|cospi| - ln|cospi/4|)`

`~= -0.34657`

Hopefully this helps!