Question

If $\int \frac{\sqrt{(x^2+1)}}{x^4}dx=-\frac{1}{k}\frac{{(1+x^2)}^{3/2}}{x^3}$. Find $k$.

Answer 1

Let $I=\int \frac{\sqrt{(x^2+1)}}{x^4}dx$

Put $x=\tan \theta \Rightarrow dx={\sec }^2\theta d\theta$

Therefore

$I=\int \frac{\sqrt{{\tan }^2\theta +1}}{{\tan }^4\theta }{\sec }^2\theta d\theta =\int \frac{\sqrt{{\sec }^2\theta }}{{\tan }^4\theta }{\sec }^2\theta d\theta$

$I=\int \frac{\sec \theta .{\sec }^2\theta }{{\tan }^4\theta }d\theta =\int \frac{\cos \theta }{{\sin }^4\theta }d\theta =-\frac{1}{3{\sin }^3\theta }=-\frac{1}{3}\frac{{(1+x^2)}^{3/2}}{x^3}$