Question

Solve $\int \frac{(x^2-1)}{(x^2+1)\sqrt{x^4+1}}dx$

Answer 1

$\int \sqrt{\frac{x-5}{x-7}}dx$

Put $\frac{x-5}{x-7}=u$

$⟹dx=\frac{1}{\frac{1}{x-7}-\frac{x-5}{(x-7{)}^2}}du$

$x=\frac{2}{u-1}+7$

Therefore $I=-2\int \frac{\sqrt{u}}{(u-1{)}^2}du$

$⟹$ Put $v=\sqrt{u}$^2}

$du=2\sqrt{u}dv$

$=2\int \frac{v^2}{(v^2–1{)}^2}dv$

$=2\int (\frac{1}{v^2-1}+\frac{1}{(v^2-1{)}^2})dv$

$=2\int \frac{dv}{v^2-1}+2\int (\frac{1}{4(v+1)}+\frac{1}{4(v+1{)}^2}-\frac{1}{4(v-1)}+\frac{1}{4(v-1{)}^2})dv$

$=2\times \frac{1}{2}log|\frac{v-1}{v+1}|+\frac{1}{2}log|v+1|-\frac{1}{2}\frac{1}{(v+1)}-\frac{1}{2}log|v-1|-\frac{1}{4(v-1)}+c$

$=log|\frac{\sqrt{\frac{x-5}{x-7}}-1}{\sqrt{\frac{x-5}{x-7}}+1}|-\frac{1}{2}log|\sqrt{\frac{x-5}{x-7}}+1|-\frac{1}{2(\sqrt{\frac{x-5}{x-7}}+1)}-\frac{1}{2}log|\sqrt{\frac{x-5}{x-7}}-1|-\frac{1}{(4\sqrt{\frac{x-5}{x-7}}+1)}+c$