Calculus Question: $\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx$

Question

How to evaluate integral $$\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx?$$ I tried substitution $u^2=x^2-1$ and $u=\sqrt{x^2-1}+x$ but it turns out too complicated. Could anyone here help me to evaluate the integral? Thanks in advance.

Answer 1

Hint; Use the Euler substitution; $$x=\frac{u^2+1}{2u}, \mathrm{d}x= \left (1- \frac{u^2+1}{2u^2}\right ) \mathrm{d}u $$ for the integrand, when you substitute back you get $$u=\sqrt{x^2-1}+x$$ The integral becomes;

$$\frac12 \int \frac{u+1}{u} \mathrm{d}u$$

which is pretty easy from now on.

Answer 2

The Euler substitution is definitely a smooth approach. But if you are not familiar with that, there is a conventional approach as well. First subtract and add 1 in the numerator so that you can split the fraction. You need to integrate $1$ and the term $\frac{1}{\sqrt{1-x^2}+x-1}$ Now multiply this term top and bottom with its conjugate: $\frac{1}{\sqrt{1-x^2}-(x-1)}$ Things will cancel in the denominator. Verify you end up with $\frac{\sqrt{1-x^2}-x+1}{2(x-1)}$. Now this fraction can be split into three seperate fractional terms. The term with the square root is a simple trig sub (I think you have seen those before) and the other two terms are even easier. This approach is not nearly as quick as the approach of UserX, but I wanted to show that with the ordinary "book"method, it can still be done

Answer 3

An other solution is to substitute x=sin(u), and then to use the well known relations $cos(u)=\frac{1-t^2}{1+t^2}$ and $sin(u)=\frac{2t}{1+t^2}$ with $t=tan(\frac{u}{2})$ to simplify the problem.

More detailed : $\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx = x + \int\frac{1}{\sqrt{x^2-1}+x-1}dx$

$\int^y \frac{1}{\sqrt{x^2-1}+x-1}dx=\int^{arcsin(y)} \frac{cos(x)}{\cos{x}+\sin{x}-1}dx$

With the two relations I gave above, we have $\frac{\cos{x}}{\cos{x}+\sin{x}-1}=1+\frac{\frac{1-t^2}{1+t^2}}{\frac{2t+1-t^2-1-t^2}{1+t^2}}=\frac{1-t^2}{t-t^2}=\frac{1+t}{t}=1+\frac{1}{t}=1+\frac{1}{\tan{\frac{u}{2}}}$

Answer 4

Could anyone here help me to evaluate the integral? Thanks in advance.

As a general rule, whenever dealing with expressions containing $\sqrt{x^2\pm1}$, the natural substitution is $x=\cosh t$, $($for $-)$, or $x=\sinh t$, $($for $+)$, since $\cosh^2t-\sinh^2t=1.~$ Then, after using the fact that $\cosh t\pm\sinh t=e^{\large\pm t}$, together with $\cosh't=\sinh t$ and $\sinh't=\cosh t,~$ let $e^t=u$, and employ partial fraction decomposition.