Question

Evaluate ∫ sec⁶x dx

Answer 1

∫ sec⁶x dx

In this integral we will use the formula,

1+tan²x = sec²x

I = ∫sec²x sec⁴x dx 

= ∫sec²x (1 + tan²x)²dx 

Now, 

Put tan x = t which means sec²xdx = dt, 

I = ∫(1+ t²)²dt 

= ∫(1+t⁴+2t²) dt 

Now put the value of t, 

which is t = tan x in above integral-

I = tanx + \(\frac{tan^5x}{5}\)+ 2.\(\frac{tan^3x}{3}\)