We can write above integral as:

----(Splitting tan⁵x)

(Using tan²x = sec²x – 1)

----(Splitting tan³x)

(Using tan²x = sec²x – 1)

Considering integral (1)

Let u = tanx

du = sec²x dx

Substituting values we get,

Substituting value of u we get,

Considering integral (2)

Let t = tanx

dt = sec²x dx

Substituting values we get,

Substituting value of t we get,

Considering integral (3)

[∵ ∫ tanx dx = -log|cosx| + C]

∴ integral becomes,

[∵ C+C+C is a constant]