Problems with integrals

Question

I am asking for help. I have a little bit of problems with the integration by substitution, in particular I do not know which substitution use when there are multiple square roots in an integral. For example I was trying to solve this: $$\int \frac{(x+1)^{1/2}-x^{1/2}}{x(x^{1/2})}\,\mathrm dx$$ I first tried with $t=(x+1)^{1/2}$, but I didn't solve it. Could someone give me some pieces of advices on how to solve this integral (I do not want you to solve this for me) or could you explain me what is the logic behind this kind of substitution? Thanks to all who will answer!!

Answer 1

To escape the roots and constant term in linear expression such as $x+1$ try substitutions like $t^2=x+1$ This will help since you can see the denominator supporting such a substitution.

Answer 2

In this case, doing $x=y^2$ and $\mathrm dx=2y\,\mathrm dy$ will make disappear two of the three square roots. You will get$$2\int\frac{\sqrt{1+y^2}-y}{y^2}\,\mathrm dy,$$which is simpler.

Answer 3

You could first try to simplify it: \begin{align} &\int\dfrac{(x+1)^{1/2}-x^{1/2}}{x(x^{1/2})}\\ =&\int\dfrac{(x+1)^{1/2}-x^{1/2}}{x^{3/2}}\\ =&\int\dfrac{(x+1)^{1/2}}{x^{3/2}}-\dfrac{x^{1/2}}{x^{3/2}}\\ =&\int\bigg(\dfrac{x+1}{x^{3}}\bigg)^3-\dfrac{1}{x}\\ =&\int\bigg(\dfrac{1}{x^{2}}+\dfrac{1}{x^{3}}\bigg)^3-\dfrac{1}{x}\\ \end{align}

Now you can expand the last bracket using the binomial expansion, and you have a sum of polynomials. No substitution needed!