Question

If f(x) = √1+x² then find its integral

Answer 1

f(x) = √(1+x²)
x = tan Ф
dx = (1+x²) dФ = sec²Ф dФ

$I= \int\limits^{}_{} {f(x)} \, dx = \int\limits^{}_{} {Sec\ \phi\ *\ Sec^2\phi} \,\ d\phi\\\\=sec\phi\ tan\phi- \int\limits^{}_{} {sec\phi\ tan\phi\ tan\phi} \, d\phi \\\\=sec\phi\ tan\phi-\int\limits^{}_{} {sec\phi\ (sec^2\phi-1)} \, d\phi \\\\=sec\phi\ tan\phi-\int\limits^{}_{} {sec^3\phi} \, d\phi+\int\limits^{}_{} {sec\phi} \, d\phi \\\\=2*I=sec\phi\ tan\phi+ln|sec\phi+tan\phi|\\\\I=\frac{1}{2}*[sec\phi\ tan\phi+ln|sec\phi+tan\phi|]$

$I= \int\limits^{}_{} {\sqrt{1+x^2} \, dx =\frac{x}{2}\sqrt{1+x^2}+\frac{1}{2}Ln|\sqrt{1+x^2}+x|}+C$