# Evaluate the following integral integrate of ((tanx)^2)((secx)^6) dx

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the trigonometric identity `sec^2 x = 1/(cos^2 x) = 1 + tan^2 x`  such that:

`int tan^2 x*sec^6 x dx = int tan^2 x(1 + tan^2 x)^2 sec^2 x dx`

You should use substitution method such that:

`tan x = t =>1/(cos^2 x) dx = dt => sec^2 x dx = dt`

Changing the variable of integrand yields:

`int tan^2 x(1 + tan^2 x)^2 sec^2 x dx = int t^2(1 + t^2)^2 dt`

You need to expand the integrand in t such that:

`int t^2(1 + t^2)^2 dt = int (t^2 + 2t^4 + t^6) dt`

You need to use the property of linearity of integral such that:

`int t^2(1 + t^2)^2 dt = int t^2 dt + int 2t^4 dt + int t^6 dt`

`int t^2(1 + t^2)^2 dt = t^3/3 + 2t^5/5 + t^7/7 + c`

Substituting back `tan x`  for t yields:

`int tan^2 x*sec^6 x dx = (tan^3 x)/3 + 2(tan^5 x)/5 + (tan^7 x)/7 + c`

Hence, evaluating the indefinite integral using trigonometric identites and substitution method yields `int tan^2 x*sec^6 x dx = (tan^3 x)/3 + 2(tan^5 x)/5 + (tan^7 x)/7 + c.`