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# How to integrate : sec^4(x/2) ?

Integrate : sec^4(x/2)

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- MadhukarLv 710 years agoFavorite Answer
∫ sec^4 (x/2) dx

= ∫ sec^2 (x/2) * sec^2 (x/2) dx

= ∫[1 + tan^2 (x/2)] * sec^2 (x/2) dx

= ∫ sec^2 (x/2) dx + ∫ tan^2 (x/2) * sec^2 (x/2) dx

= 2tan(x/2) + ∫ tan^2 (x/2) * sec^2 (x/2) dx

Let tan(x/2) = u for the second integral

=> (1/2) sec^2 (x/2) dx = du

=> sec^2 (x/2) dx = 2du

=> Integral

= 2tan(x/2) + 2 ∫ u^2 du

= 2tan(x/2) + (2/3) u^3 + c

= 2tan(x/2) + (2/3) tan^3 (x/2) + c.

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