Question

∫ tan⁴ (½)z dz

Answer 1

$∫tan^4⁡(1/2 z) dz$

let $u=1/2 z ⇒ du= 1/2 dz or 2du=dz$

$؞ ∫tan^4⁡(1/2 z) dz=2∫tan^4⁡ udu …(i)$

we define

$ǀ_n= ∫tan^n⁡ udu= (tan^{n-1} u)/(n-1) - ǀ_(n-1) for n≥2$

as the reduction formula

thus,

$ǀ_4= ∫tan^4⁡udu= tan^3⁡u /3- ǀ_2 …(ii)$

$ǀ_2 = ∫tan^2⁡u du= tan⁡u- ǀ_0 …(iii)$

$ǀ_0 = ∫du=u …(iv)$

replacing (iv) in (iii) gives

$ǀ_(2 )=tan⁡ u-u …(v)$

replacing (v) in (ii) gives

$ǀ_4 = tan^3⁡u /3-(tan⁡u-u)$

$tan^3⁡u /3-tan⁡u -u …(vi)$

replacing (vi) in (i) gives

$∫tan^4⁡(1/2 z)dz =2[tan^3⁡(1/2 z)/3-tan⁡(1/2 z)+ 1/2 z] +c$

$=2/3 tan^3⁡(1/2 z )-2tan⁡(1/2 z)+z+c$