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

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

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

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

- x + \frac{π}{3} + tan (x) - 3

Solution steps

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

Rewrite using trig identities

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

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

\int_{\frac{π}{3}}^{x} sec^{2} (t) d t = tan (x) - 3

= - (x - \frac{π}{3}) + tan (x) - 3

Simplify

= - x + \frac{π}{3} + tan (x) - 3

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