Question

`int (cos x + x sin x)/(x(x + cos x))`dx = ?

Options

• `log |1/(x + cos x)| + c`

• `log |x/(x + cos x)| + c`

• `log |1/(x + sin x)| + c`

• `log |x/(x + sin x)| + c`

Answer 1

`log |x/(x + cos x)| + c`

Explanation:

`int (cos x + x sin x)/(x(x + cos x))`dx

`= int ((x + cos x) - (x - x sin x))/(x(x + cos x))`dx

`= int (x + cos x)/(x(x + cos x)) "dx" - int (x (1 - sin x))/(x (x + cos x))`dx

`= int "dx"/x - int(1 - sin x)/(x + cos x) "dx"`

= log |x| - log |x + cos x| + c   ....`[because int ("f'"(x))/("f"(x)) "d"x = log |"f"(x)| + c]`

`log |x/(x + cos x)| + c`