Sunday Afternoon Maths LIII

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$$111...1=(10n\pm1)^2$$$$=100n^2\pm20n+1$$ $$=10(10n^2\pm2n)+1$$

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$$\int_0^{\frac\pi2}\frac1{1+\cot^a(x)}\,dx?$$

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$$I_1=\int_0^{\frac\pi2}\frac1{1+\tan^a(x)}\,dx $$$$=\int_0^{\frac\pi2}\frac{\cos^a(x)}{\cos^a(x)+\sin^a(x)}\,dx$$ $$I_2=\int_0^{\frac\pi2}\frac1{1+\cot^a(x)}\,dx $$$$=\int_0^{\frac\pi2}\frac{\sin^a(x)}{\cos^a(x)+\sin^a(x)}\,dx$$ $$I_1+I_2 = \int_0^{\frac\pi2}\frac{\cos^a(x)+\sin^a(x)}{\cos^a(x)+\sin^a(x)}\,dx$$$$=\int_0^{\frac\pi2}1\,dx$$$$=\frac\pi2$$

Extension

$$\int_0^{\frac\pi2}\frac1{1-\tan^a(x)}\,dx?$$