Question

integration class 12th ​

Answer 1

Multiply numerator and denominator by x⁻². We'll get

$I = \int\limits {\frac{1-x^{-2}}{x^2+x^{-2}} } \, dx$

$=\int\limits {\frac{1-x^{-2}}{x^2+x^{-2}+2-2} } \, dx$

$= \int\limits {\frac{1-x^{-2}}{(x+x^{-1})^2 -2} } \, dx$

Take t = x + x⁻¹

dt = (1-x⁻²)dx

So,

$I= \int\limits {\frac{1}{t^2-2}} } \, dt$

Applying the formula for $\int\limits {\frac{1}{x^2-a^2} } \, dx$, we get

$I = \frac{1}{2\sqrt{2} }log|\frac{t-\sqrt{2} }{t+\sqrt{2} }|$

$I = \frac{1}{2\sqrt{2} }log|\frac{x+x^{-1}-\sqrt{2} }{x+x^{-1}+\sqrt{2} }|+c$