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

$$find{\int }_0^{\frac{\pi }{4}}\frac{{cos}^{5}t}{cos\left(5t\right)}dt$$

Answered by bobhans last updated on 29/Mar/21

$$\cos \left(5\mathrm{t}\right)=\cos {}^5\mathrm{t}-10\sin {}^2\mathrm{t}\cos {}^3\mathrm{t}+5\sin {}^4\mathrm{t}\cos \mathrm{t}$$$$\cos \left(5\mathrm{t}\right)=\cos {}^5\mathrm{t}-10\cos {}^3\mathrm{t}(1-\cos {}^2\mathrm{t})+5\cos \mathrm{t}{(1-\cos {}^2\mathrm{t})}^2$$$$=\cos {}^5\mathrm{t}-10\cos {}^3\mathrm{t}+10\cos {}^5\mathrm{t}+5\cos \mathrm{t}(1-2\cos {}^2\mathrm{t}+\cos {}^4\mathrm{t})$$$$=11\cos {}^5\mathrm{t}-10\cos {}^3\mathrm{t}+5\cos \mathrm{t}-10\cos {}^3\mathrm{t}+5\cos {}^5\mathrm{t}$$$$=16\cos {}^5\mathrm{t}-20\cos {}^3\mathrm{t}+5\cos \mathrm{t}$$$$\mathrm{I}=\int \frac{\cos {}^4\mathrm{t}}{16\cos {}^4-20\cos {}^2\mathrm{t}+5}\mathrm{dt}$$$$$$

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