Question

16. tan4x

Answer 1

The given function is tan 4 x ,

∫ tan 4 xdx = ∫ tan 2 x tan 2 xdx (1)

From (1),

tan 4 x=( sec 2 x−1 ) tan 2 x = sec 2 x tan 2 x− tan 2 x = sec 2 x tan 2 x− sec 2 x+1 (2)

From (1) and (2), we get,

∫ tan 4 xdx = ∫ ( sec 2 x tan 2 x− sec 2 x+1 )dx = ∫ ( sec 2 x tan 2 x ) dx− ∫ sec 2 xdx+ ∫ 1 dx = ∫ ( sec 2 x tan 2 x ) dx−tanx+x+c (3)

Consider, ∫ ( sec 2 x tan 2 x ) dx(4)

Let, tanx=t

Differentiate with respect to x,

sec 2 xdx=dt

Substitute value of t and dt in (4),

∫ ( sec 2 x tan 2 x ) dx= ∫ t 2 dt = t 3 3 +c = tan 3 x 3 +c

Combine results of (3) and (4),

∫ tan 4 xdx = tan 3 x 3 −tanx+x+c

Thus, the integral of the function tan 4 x is tan 3 x 3 −tanx+x+c.