Question

how to integrate log(sec x)

Answer 1

Answer:

$\large\boxed{\sf{ x log( \sec(x) ) - ( \frac{ {x}^{3} }{3} + \frac{ {x}^{5} }{15} + \frac{ 2{x}^{7} }{105} + ..........) + c}}$

Step-by-step explanation:

$\displaystyle \int log( \sec(x) ) dx\\ \\ = log( \sec(x) ) \displaystyle \int dx - \displaystyle \int ( \frac{d}{dx} log( \sec(x) ) ) \displaystyle \int dx)dx \\ \\ = x log( \sec(x) ) - \displaystyle \int x \tan(x) dx \\ \\ = x log( \sec(x) ) - x\displaystyle \int (x + \frac{ {x}^{3} }{3} + \frac{2 {x}^{5} }{15} + ........)dx \\ \\ = x log( \sec(x) ) - \displaystyle \int ( {x}^{2} + \frac{ {x}^{4} }{3} + \frac{2 {x}^{6} }{15} + .........)dx \\ \\ = x log( \sec(x) ) - ( \frac{ {x}^{3} }{3} + \frac{ {x}^{5} }{15} + \frac{ 2{x}^{7} }{105} + ..........) + c$

We have used the integration by product rule and the expansion of tan x.

Answer 2

Answer:

see the attachment above

Step-by-step explanation:

answer turn out to be zero

hence this cant be integrated