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Question Number 61165 by Tawa1 last updated on 29/May/19

Answered by MJS last updated on 30/May/19

$$\int \frac{{\cos }^2(2x-5)\cos (2x-14)}{\cos (2x-7)}dx=$$$$[t=2x-7\to dx=\frac{dt}{2}]$$$$=\frac{1}{2}\int \frac{{\cos }^2(t+2)\cos (t-7)}{\cos \left(t\right)}dt=$$$$[\mathrm{use}\mathrm{these}:$$$${\cos }^2(t+2)={(\cos 2\cos t-\sin 2\sin t)}^2$$$$\cos (t-7)=\cos 7\cos t+\sin 7\sin t$$$$\mathrm{let}\mathrm{me}\mathrm{write}{c}_{\alpha },s_{\alpha }\mathrm{for}\cos \alpha ,\sin \alpha ]$$$$=\frac{c_2^2{c}_7}{2}\int {\cos }^{2}tdt+\frac{c_2(c_2{s}_7-2c_7{s}_2)}{2}\int \cos t\sin tdt+\frac{s_2(c_7{s}_2-2c_2{s}_7)}{2}\int {\sin }^{2}tdt+\frac{s_2^2{s}_7}{2}\int {\sin }^{2}t\tan tdt$$$$...\mathrm{now}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{easy}\mathrm{to}\mathrm{solve}:$$$$\int {\cos }^{2}tdt=\frac{1}{2}\int (1+\cos 2t)dt=\frac{1}{2}t+\frac{1}{4}\sin 2t$$$$\int \cos t\sin tdt=\frac{1}{2}\int \sin 2tdt=-\frac{1}{4}\cos 2t$$$$\int {\sin }^{2}tdt=\frac{1}{2}\int (1-\cos 2t)dt=\frac{1}{2}t-\frac{1}{4}\sin 2t$$$$\int {\sin }^{2}t\tan tdt=\int (\tan t-\cos t\sin t)dt=$$$$=\int \tan tdt-\int \cos t\sin tdt=-\ln \cos t+\frac{1}{4}\cos 2t$$$$\mathrm{now}\mathrm{re}-\mathrm{substitute}t=2x-7\mathrm{and}\mathrm{finally}$$$$\mathrm{add}\mathrm{the}\mathrm{constant}\mathrm{factors}\mathrm{and}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{done}$$

Commented by Tawa1 last updated on 30/May/19

$$\mathrm{Ohh}.\mathrm{I}\mathrm{just}\mathrm{saw}\mathrm{it}\mathrm{now}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$$$\mathrm{I}\mathrm{will}\mathrm{appreciate}\mathrm{if}\mathrm{you}\mathrm{can}\mathrm{help}\mathrm{me}\mathrm{finish}\mathrm{up}\mathrm{sir}.$$$$\mathrm{When}\mathrm{you}\mathrm{are}\mathrm{less}\mathrm{busy},\mathrm{i}\mathrm{will}\mathrm{study}\mathrm{it}.\mathrm{Thanks}\mathrm{sir}\mathrm{for}\mathrm{your}\mathrm{help}.$$

Commented by MJS last updated on 30/May/19

$$\mathrm{did}\mathrm{a}\mathrm{few}\mathrm{steps}\mathrm{more}...$$

Commented by Tawa1 last updated on 30/May/19

$$\mathrm{Yes},\mathrm{i}\mathrm{get}\mathrm{it}\mathrm{now}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{more}.\mathrm{I}\mathrm{appreciate}.$$

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