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For all x in (-pi/2, pi/2), let f(x) = 7*tan⁸(x) + 7*tan⁶(x) - 3*tan⁴(x) - 3*tan²(x). Which of the following are correct?

1. Integral from 0 to pi/4 of x*f(x) dx = 1/12
2. Integral from 0 to pi/4 of f(x) dx = 0
3. Integral from 0 to pi/4 of x*f(x) dx = 1/6
4. Integral from 0 to pi/4 of f(x) dx = 1

Correct answer: Integral from 0 to pi/4 of x*f(x) dx = 1/12

Solution

After substitution t = tan x, the integral of f from 0 to pi/4 becomes integral of (7t⁶ - 3t²) dt from 0 to 1 = [t⁷ - t³] from 0 to 1 = 1 - 1 = 0. For the x*f(x) integral, use integration by parts.

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