Integration Questions

Question Number 10241 by FilupSmith last updated on 31/Jan/17

$$\int_{{a}} ^{\:{a}+\delta} \frac{\mathrm{sin}\left({x}\right)}{\mathrm{cos}\left({x}+\delta\right)}{dx}\:=\:??? \\$$

Commented by prakash jain last updated on 31/Jan/17

$$\frac{\mathrm{sin}\left({x}\right)}{\mathrm{cos}\left({x}+\delta\right)}=\frac{\mathrm{sin}\left({x}+\delta−\delta\right)}{\mathrm{cos}\left({x}+\delta\right)} \\$$$$=\frac{\mathrm{sin}\:\left({x}+\delta\right)\mathrm{cos}\:\delta−\mathrm{cos}\:\left({x}+\delta\right)\mathrm{sin}\:\delta}{\mathrm{cos}\:\left({x}+\delta\right)} \\$$$$=\mathrm{tan}\:\left({x}+\delta\right)\mathrm{cos}\:\delta−\mathrm{sin}\:\delta \\$$$$\int_{{a}} ^{{a}+\delta} \left(\mathrm{tan}\:\left({x}+\delta\right)\mathrm{cos}\:\delta−\mathrm{sin}\:\delta\right){dx} \\$$$$=\left[−\mathrm{cos}\:\delta\mathrm{ln}\:\mathrm{cos}\:\left({x}+\delta\right)−{x}\mathrm{sin}\:\delta\right]_{{a}} ^{{a}+\delta} \\$$$$=\left[−\mathrm{cos}\:\delta\mathrm{ln}\:\frac{\mathrm{cos}\:\left({a}+\delta\right)}{\mathrm{cos}\:{a}}−\delta\mathrm{sin}\:\delta\right] \\$$