Question

How do you evaluate the definite integral `intsec^2(t/4)` from `[0, pi]`?

Answer 1

`int_0^pi sec^2(t/4)dt = 4` units

Explanation:

This is a relatively simple integral because the antiderivative of `sec^2(t)` is `tan(t)` (the derivative of tangent is secant squared; we just work in reverse when integrating). The only issue is that `t/4` within the secant function. To fix it, we reverse the chain rule - we multiply the result by 4. So:
`int_0^pi sec^2(t/4)dt = [4tan(t/4)]_0^pi`

Evaluating this from `0` to `pi`:
`=(4tan((pi)/4)-4tan((0)/4))`
`=(4(1)-4(0))`
`=4`

And that's it.