u = x^2 + 1, du/dx = 2x, du/2x = dx
[tex] \int\limits_ {} \frac{1}{ x^{2}+1} \, dx = \int\limits { \frac{1}{u} \frac{du}{2x} }
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[tex]x = tan 0
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[tex] \frac{dx}{d0} =sec^{2}0
dx=sec ^{2}0 d 0
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[tex] x^{2} +1 =( tan 0^{2} ) +1 = tan^{2}0 +1 =sec 0
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[tex] \int\limits { \frac{1}{ x^{2}+1} } \, dx = \int\limits { \frac{1}{sec0}( sec^{2}0 d0) } = \int\limits{sec0 d0} \, = ln(tan 0 +sec 0) +C =
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[tex]= ln[x+( x^{2} +1)]+C = ln ( x^{2} +x+1)[/tex]
