Is this integral can even be solved?

by Soumalya Pramanik   Last Updated May 16, 2019 06:20 AM

As I need to study calculus for Physics in Grade 11, I learnt quite a bit and I can say I'm familiar with differentiation and Basic Integrals. But As I'm too curious for my own good, I decided to try out $\mathbf {Integration}$ $\mathbf{By}$ $\mathbf{Parts}$. I solved some easy ones. Then I thought, let's solve $$\int{x \sec x dx}$$

As I tried to solve it, I came up with this :

$$\int{x \sec x dx}$$
$$=x \int \sec x dx - \int \frac d{dx} (x) \left(\int \sec xdx\right)dx$$ $$=x \ln|\sec x + \tan x| - \int \ln |\sec x + \tan x| dx$$ $$=x \ln|\sec x + \tan x| - \ln|\sec x + \tan x| \int dx + \int \frac d{dx}(\ln|\sec x + \tan x|) (\int \ln|\sec x +\tan x|dx)dx$$ $$= x\ln |\sec x + \tan x| - x\ln|\sec x + \tan x| + \int \sec x (\int \ln |\sec x + \tan x| dx)dx$$

And it goes on, quite annoyingly.

I'm new to this, so I might have made some fundamental mistakes, but I still don't think this can be solved. A Mathematics book written in my vernacular informed me that some integral expressions cannot be simplified, so I guess this expression can be like it. Kindly look into this and let me know anything about it. Thanks in advance!

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