Evaluate Indefinite Integral

Use the given trigonometric identity to set up a u-substitution and then evaluate the indefinite integral of sec^4x dx...please explain step by step...thank you :)

sec(x)^4 * dx =>
(sec(x)^2)^2 * dx =>
sec(x)^2 * sec(x)^2 * dx =>
(1 + tan(x)^2) * sec(x)^2 * dx =>
sec(x)^2 * dx + tan(x)^2 * sec(x)^2 * dx

u = tan(x)
du = sec(x)^2 * dx

du + u^2 * du

Now integrate:

u + (1/3) * u^3 + C =>
(1/3) * u * (3 + u^2) + C =>
(1/3) * tan(x) * (3 + tan(x)^2) + C

... ∫ ( sec⁴ x ) dx

= ∫ ( sec² x ) · sec² x dx

= ∫ ( 1 + tan² x )· sec² x dx

= ∫ ( 1 + u² ) du, ................... u = tan x, du = sec² x dx

= u + ( u³ / 3 ) + C

= ( sec x ) + (1/3)· sec³ x + C ..................... Ans.
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