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Question Number 25777 by abhishekkumar22121999@gmail.co last updated on 14/Dec/17

∫(1/(sin x+cos x))dx

$$\int\frac{\mathrm{1}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{dx} \\ $$

Answered by mrW1 last updated on 14/Dec/17

=∫(1/((√2)(sin x cos (π/4)+sin (π/4) cos x))) dx  =∫(1/((√2) sin (x+(π/4)))) dx  =(1/(2(√2)))×ln ((1−cos (x+(π/4)))/(1+cos (x+(π/4))))+C

$$=\int\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\left(\mathrm{sin}\:{x}\:\mathrm{cos}\:\frac{\pi}{\mathrm{4}}+\mathrm{sin}\:\frac{\pi}{\mathrm{4}}\:\mathrm{cos}\:{x}\right)}\:{dx} \\ $$$$=\int\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)}\:{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}×\mathrm{ln}\:\frac{\mathrm{1}−\mathrm{cos}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)}{\mathrm{1}+\mathrm{cos}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)}+{C} \\ $$

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