I = ∫ x 2 sec x 3 d x Let, x 3 = t ⇒ 3 x 2 d x = d t ⇒ x 2 d x = 1 3 d t ∴ I = ∫ sec t · 1 3 d t = 1 3 log e sec t + tan t + C = 1 3 log e sec x 3 + tan x 3 + C , where C is constant of integration.

∫ x 1 3 sin x 4 3 d x

I = ∫ x 1 3 sin x 4 3 d x Let x 4 3 = t ⇒ 4 3 x 4 3 - 1 d x = d t ⇒ x 1 3 d x = 3 4 d t ∴ I = ∫ sin t · 3 4 d t = - 3 4 cos t + C = - 3 4 cos x 4 3 + C , where C is constant of integration.

∫ tan x sec 2 x x d x

∫ tan x sec 2 x x d x Let, tan x = t ⇒ sec 2 x · 1 2 x d x = d t ⇒ sec 2 x x d x = 2 d t ∴ I = 2 ∫ t d t = 2 t 2 2 + C = tan 2 x + C , where C is constant of integration.

∫ sec 2 x + 1 x + 1 d x

I = ∫ sec 2 x + 1 x + 1 d x Let, x + 1 = t ⇒ 1 2 x + 1 d x = d t ⇒ d x x + 1 = 2 d t ∴ I = ∫ sec 2 t · 2 d t = 2 tan t + C = 2 tan x + 1 + C , where C is constant of integration.