If-g-x-0-x-cos-4-t-dt-then-g-x-pi-

Question Number 30354 by soksan last updated on 21/Feb/18

If g (x) = {\int^{x}}_{0} \cos^{4} t d t, then g (x + π) =

Answered by ajfour last updated on 21/Feb/18

g (x) = \frac{1}{4} \int_{0}^{x} {(2 \cos^{2} t)}^{2} d t

= \frac{1}{4} \int_{0}^{x} {(1 + \cos 2 t)}^{2} d t

= \frac{1}{4} \int_{0}^{x} (1 + 2 \cos 2 t + \cos^{2} 2 t) d t

= \frac{x}{4} + \frac{\sin 2 x}{4} + \frac{1}{8} \int_{0}^{x} (1 + \cos 4 t) d t

= \frac{x}{4} + \frac{\sin 2 x}{4} + \frac{x}{8} + \frac{\sin 4 x}{32}

g (x + π) = g (x) + \frac{3 π}{8} .

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