Question

`int1/(sqrt(1 + cos 2x)) dx` is equal to ______.

Options

• log cos x + C

• `1/sqrt2` log |sec x + tan x| + C

• `1/sqrt2` log |sec x − tan x| + C

• log sin 2x + C

Answer 1

`int1/(sqrt(1 + cos 2x)) dx` is equal to `bbunderline(1/sqrt2)` log |sec x + tan x| + C.

Explanation:

Simplify the expression using identities:

cos 2x = 2 cos² x − 1

= 1 + cos 2x

= 2 cos² x

Substitute this into the integral:

`int 1/sqrt(2 cos^2 x) dx`

Simplify the square root:

`int 1/(sqrt2 . cos x) dx`

Assuming cos x is positive in the interval of integration:

`int 1/(sqrt2 . cos x) dx = 1/sqrt2 int sec x  dx`

The standard integral for sec x is log |sex x + tan x| + C.

Substituting this back:

`1/sqrt2 log |sec x + tan x| + C`