What is the integral from pi/3 to x of cot^2(t) ?

The integral from pi/3 to x of cot^2(t) is -x+pi/3+1/(sqrt(3))-cot(x)

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integral from pi/3 to x of cot^2(t)

\int_{\frac{π}{3}}^{x} cot^{2} (t) d t

- x + \frac{π}{3} + \frac{1}{3} - cot (x)

Solution steps

\int_{\frac{π}{3}}^{x} cot^{2} (t) d t

Rewrite using trig identities

= \int_{\frac{π}{3}}^{x} - 1 + csc^{2} (t) d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{\frac{π}{3}}^{x} 1 d t + \int_{\frac{π}{3}}^{x} csc^{2} (t) d t

\int_{\frac{π}{3}}^{x} 1 d t = x - \frac{π}{3}

\int_{\frac{π}{3}}^{x} csc^{2} (t) d t = - cot (x) + \frac{1}{3}

= - (x - \frac{π}{3}) - cot (x) + \frac{1}{3}

Simplify

= - x + \frac{π}{3} + \frac{1}{3} - cot (x)

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What is the integral from pi/3 to x of cot^2(t) ?
The integral from pi/3 to x of cot^2(t) is -x+pi/3+1/(sqrt(3))-cot(x)