Question

The integral `intcos(log_(e)x)dx` is equal to: (where C is a constant of integration)
A. `(x)/(2)[cos(log_(e)x)+sin(log_(e)x)]+C`
B. `x[cos(log_(e)x)+sin(log_(e)x)]+C`
C. `x[cos(log_(e)x)-sin(log_(e)x)]+C`
D. `(x)/(2)[cos(log_(e)x)-sin(log_(e)x)]+C`

Answer 1

Correct Answer - A
Let `I=int cos(log_(e)x)dx`
`=x cos(log_(e)x)-intx(-sin(log_(e)x))(1)/(x)*dx" "["using integration by parts"]`
`= x cos(log_(e)x)+int sin(log_(e)x)dx`
`=x cos(log_(e)x)+ x sin(log_(e)x)-int x(cos(log_(e)x))(1)/(x)dx" "["again, using integration by parts"]`
`rArr I = x cos(log_(e)x)+x sin(log_(e)x)-I`
`rArr I=(x)/(2)[cos(log_(e)x)+sin(log_(e)x)]+C`.