Please help with integral proof problem

[miniskus]

First i need to find the int[{(tanx)^3}*{(secx)^4}dx in two different ways and then i have to prove that my solutions are equal. The first way I used i got as being (1/6)[(tan(x))^6] + (1/4)[(tan(x))^4] and the second as (1/6)[(sec(x))^6] - (1/4)[(sec(x))^4)], but these aren't exactly equal so i don't know what I'm doing wrong. Thank you very much for your help.

 

[soroban]

Hello, miniskus!

What you got is absolutely correct . . . nice work!

Of course, the answers look differerent.
Our task is to show that they are indeed equal (like an identity).

Let's start with the secant-answer. (Don't forget the "plus C".)
. . . (1/6)sec⁶x - (1/4)sec⁴x + C

Factor out 1/12, and factor out sec⁴x: . (1/12)sec⁴x (2sec²x - 3) + C

So we have: . (1/12) (sec²x)² (2sec²x - 3) + C

Use the identity: .sec²x .= .tan²x + 1 . and substitute.
. . . (1/12) (tan²x + 1)² (2[tan²x + 1) - 3) + C

Multiply it out and simplify: . (1/12)(2tan⁶x + 3tan⁴x - 1) + C

This is: . (1/6)tan⁶x + (1/4)tan⁴x - 1/12 + C

. . which is your other answer (differing only by a constant).