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What is the integration of log cos x dx ?

Solution :

We have, I = log cos x dx

By using integraton by parts,

I = 1.log cos x dx

Taking log cos x as first function and 1 as second function. Then,

I = log cos x 1 dx – { ddx (log cos x) 1 dx } dx

I = x log cos x – { sinxcosxx } dx

I = x log cos x + x tan x dx

Again using integration by parts,

I = x log cos x + x log |sec x | – {1.(log sec x)} dx

I = x log cos x + x log sec x – I

2I = x log cos x + x log sec x + C

I = xlogcosx2 + xlogsecx2 + C

Hence, the integration of log cos x with respect to x is xlogcosx2 + xlogsecx2 + C


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