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Question Number 118094 by bemath last updated on 15/Oct/20

$$\int \frac{\mathrm{dx}}{\sec \mathrm{x}+2}=?$$

Answered by Dwaipayan Shikari last updated on 15/Oct/20

$$\frac{1}{2}\int \frac{2cosxdx}{2cosx+1}=\frac{1}{2}\int 1-\frac{1}{2cosx+1}dx$$$$=\frac{x}{2}-\int \frac{1}{\frac{2-2t^2}{1+t^2}+1}.\frac{1}{1+t^2}t=tan\frac{x}{2}$$$$=\frac{x}{2}-\int \frac{1}{3-t^2}dt$$$$=\frac{x}{2}+\frac{1}{2\sqrt{3}}log\left(\frac{t-\sqrt{3}}{t+\sqrt{3}}\right)=\frac{x}{2}+\frac{1}{2\sqrt{3}}log\left(\frac{tan\frac{x}{2}-\sqrt{3}}{tan\frac{x}{2}+\sqrt{3}}\right)+C$$

Commented by bemath last updated on 15/Oct/20

$$\mathrm{weirstrass}\mathrm{substitution}$$$$\mathrm{santuy}...$$

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