Evaluate the integral cos⁷3x dx with the limits from 0 to π/6.
"\\int _0^{\\frac{\\pi }{2}}\\cos ^7\\left(3x\\right)\\frac{1}{3}d3x"
"=\\frac{1}{3}\\cdot \\int _0^{\\frac{\\pi }{2}}\\cos ^7\\left(3x\\right)d3x"
"=\\frac{1}{3}\\cdot \\int _0^{\\frac{\\pi }{2}}\\left(1-\\sin ^2\\left(3x\\right)\\right)^3\\cos \\left(3x\\right)d3x"
"=\\frac{1}{3}\\cdot \\int _0^1\\left(1-v^2\\right)^3dv"
"=\\frac{1}{3}\\cdot \\int _0^11-3v^2+3v^4-v^6dv"
"=\\frac{1}{3}\\left(\\int _0^11dv-\\int _0^13v^2dv+\\int _0^13v^4dv-\\int _0^1v^6dv\\right)"
"=\\frac{1}{3}\\left(1-1+\\frac{3}{5}-\\frac{1}{7}\\right)"
"=\\frac{16}{105}"
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