Question

\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]

Answer 1

 

` ∫  {x  dx}/{\sqrt{x^4 + a^4}} `

 ` ∫  {x  dx}/\sqrt{(x^2)^2 + (a^2)^2}`
` \text{  let}   x^2 = t `
\[ \Rightarrow\text{  2x dx } = dt\]
\[ \Rightarrow\text{  x dx } = \frac{dt}{2}\]
Now,  ` ∫  {x  dx}/\sqrt{(x^2)^2 + (a^2)^2}`
\[ = \frac{1}{2}\int\frac{dt}{\sqrt{t^2 + \left( a^2 \right)^2}}\]
\[ = \frac{1}{2} \text{ log }\left| t + \sqrt{t^2 + a^4} \right| + C\]
\[ = \frac{1}{2} \text{ log }\left| x^2 + \sqrt{x^4 + a^4} \right| + C\]