Integral of trig functions.. :(

how do you solve the
integral of x(csc^(2)x)dx from pi/4 to pi/2

int: x * (csc^2(x)) dx

Using integration by parts, let:

u = x
du = dx
dv = csc^2(x) dx
v = -cot(x)

This v can be found with some simple trig substitution / knowing your trig rules, I'll let you confirm this on your own! From here, we have:

uv - int: vdu as:

-x * cot(x) - int: -cot(x) dx

Pull out the negative:

-x * cot(x) + int: cot(x) dx

The integral of cot(x) (again using some trig integration) is ln(sin(x)), so we have:

-x * cot(x) + ln(sin(x))

Re-arrange, and we have the indefinite integral of:

ln(sin(x)) - x * cot(x)

Now, we simply plug in our bounds (pi/4, pi/2):

(ln(sin(pi/2)) - (pi/2) * cot(pi/2)) - (ln(sin(pi/4)) - (pi/4) * cot(pi/4))

Simplify with a calculator, and we have the definite integral:

(pi/4) + (ln(2) / 2)

Since cot(x) = 1/tan(x), you would use the tangent button but then do 1 divided by the answer. Good luck!

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