What is the indefinate integral

What have you tried so far?

 

Basically, I believe you must simply save a factor of [tex] \sec ^ 2 x[/tex]

and use [tex]\sec ^ 2 x = 1 + \tan ^ 2 x [/tex] to express the remaining factors

in terms of [itex] \tan x [/itex]. Next, simply substitute with respect to [tex] \tan x [/tex].

(I think it's called "u"-substitution in some texts).?

As you should already know, :shy:
[tex] \frac{d}{dx} \tan x = \sec ^ 2 x [/tex]

*Here's that method, put in action :
[tex] \int {\tan ^7 x\sec ^4 x\,dx} = \int {\tan ^7 x\left( {1 + \tan ^2 x} \right)\sec ^2 x\,dx} = [/tex]
[tex] \int {\tan ^7 x\left( {1 + \tan ^2 x} \right)\,d\left( {\tan x} \right)} = \boxed{\frac{{\tan ^8 x}}{8} + \frac{{\tan ^{10} x}}{{10}} + C} [/tex]

!It is very likely that somewhere I made an error!....
Somewhere...some silly error

 

No, this is correct