find-0-pi-4-cos-5-t-cos-5t-dt-

Question Number 137003 by Mathspace last updated on 28/Mar/21

f i n d \int_{0}^{\frac{π}{4}} \frac{{c o s}^{5} t}{c o s (5 t)} d t

Answered by bobhans last updated on 29/Mar/21

\cos (5 t) = \cos^{5} t - 10 \sin^{2} t \cos^{3} t + 5 \sin^{4} t \cos t

\cos (5 t) = \cos^{5} t - 10 \cos^{3} t (1 - \cos^{2} t) + 5 \cos t {(1 - \cos^{2} t)}^{2}

= \cos^{5} t - 10 \cos^{3} t + 10 \cos^{5} t + 5 \cos t (1 - 2 \cos^{2} t + \cos^{4} t)

= 11 \cos^{5} t - 10 \cos^{3} t + 5 \cos t - 10 \cos^{3} t + 5 \cos^{5} t

= 16 \cos^{5} t - 20 \cos^{3} t + 5 \cos t

I = \int \frac{\cos^{4} t}{16 \cos^{4} - 20 \cos^{2} t + 5} dt

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