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Question Number 86365 by Prithwish Sen 1 last updated on 28/Mar/20

$$\mathrm{I}\mathrm{think}\mathrm{it}\mathrm{will}\mathrm{be}$$$${\int }_0^{\frac{\pi }{4}}\frac{\mathrm{dx}}{\sqrt{1+\mathrm{tanx}}}\approx {\int }_0^{\frac{\pi }{4}}\frac{\mathrm{dx}}{\sqrt{1+\mathrm{x}}}$$$$=\frac{\mathit{\pi }}{4}-\frac{1}{2}.\frac{1}{2}.{\left(\frac{\mathit{\pi }}{4}\right)}^2+\frac{1.3}{2.4}.\frac{1}{3}.{\left(\frac{\mathit{\pi }}{4}\right)}^3-\frac{1.3.5}{2.4.6}.\frac{1}{4}{\left(\frac{\mathit{\pi }}{4}\right)}^4+....$$

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