Compute $\int_0^1\frac{\arctan(\lambda x)}{x\sqrt{1-x^2}}\mathrm dx$ and $\int_0^1\frac{\arctan(\lambda x)}{\sqrt{1-x^2}}\mathrm dx$ A friend gave me this integration as a challenge that he had failed to solve. I tried substituting $\tan^{-1}(\lambda x)=\theta$. I also tried integrating by parts. None worked, any advice/ hints would be greatly appreciated. Even better if someone could tell me what is a general way to approach problems like these!
$$\int_{0}^{1}\frac{\tan^{-1}(\lambda x)}{x\sqrt{1-x^{2}}}dx$$
The second part of the problem asks the value of
$$\int_{0}^{1}\frac{\tan^{-1}(\lambda x)}{\sqrt{1-x^{2}}}dx$$
 A: I see that you only asked for advice and hints. Hence I will not give all the steps unless you request them.
First, the solution
$$
\int_0^1\frac{\text{arctan}(\lambda x)}{x\sqrt{1-x^2}}\text{d}x = \frac{\pi}{2}\text{arcsinh}(\lambda)
$$
As suggested in the comments, the way to arrive at this expression is to "differentiate wrt to $\lambda$" also known as the Feynman's trick. The steps are the following.

*

*First, define $f(x,\lambda) = \frac{\text{arctan}(\lambda x)}{x\sqrt{1-x^2}}$ and $F(\lambda) = \int_0^1\frac{\text{arctan}(\lambda x)}{x\sqrt{1-x^2}}\text{d}x = \int_0^1f(x,\lambda)\text{d}x$.


*Second, note that $F(0) = 0$ since $\text{arctan}(0)=0$.


*Now, compute $\frac{\partial f}{\partial\lambda}$. In this step, instead of the $\text{arctan}{\lambda x}$ term you will get a rational function.


*Now, note that $\frac{dF}{d\lambda} = \int_0^1\frac{\partial f}{\partial\lambda}dx$ which is an integral that should be easier to compute. Just be careful for this particular case when substituting the limits of integration. You may need to compute the (one-sided) limit $x\to 1^-$ instead of just plugging in $x=1$.


*Now that you have $\frac{dF}{d\lambda}$ just integrate $F(\lambda)=\int \frac{dF}{d\lambda} d\lambda + C$ and use the fact that $F(0)=0$ to compute the integration constant $C$.
After all these, you should be able to arrive at $(\pi/2)\text{arcsinh}(\lambda)$
If you want to spoil yourself, these are the sequence of computations I did in wolfram following this procedure to obtain the result. Click them only of you get stuck again:
Step 3,
Step 4: anti derivative,
Step 4: limit,
Step 5
Regarding the second integral you asked for, I think I can confirm that the procedure is the same. Some of the computations you made for the first integral will be useful.
