Integrate the following: $\int {\sqrt {1 + \cos 2x} {\text{ dx}}} $

Hint: Simplify the integrand using the half angle formula and then integrate it.

Complete step-by-step answer:

We have to integrate,

$\int {\sqrt {1 + \cos 2x} {\text{ dx}}} $

Using the half angle formula, $\left( {1 + \cos 2x = 2{{\cos }^2}x} \right) \\$

So, the integral now changes to $\int {\sqrt {2{{\cos }^2}x{\text{ }}} {\text{ }}} dx \\$

Now $\sqrt {2}$, being a constant, can be taken out of integral sign,

$I = \sqrt 2 \int {\cos x{\text{ dx}}} \\$

Integration of cosx is sinx, we get

$I = {\text{ }}\sqrt 2 {\text{ sinx}} + C\\$

where, C is the integration constant.

Note: Usage of the half angle formula makes the solution easier. Students often forget to add the integration constant which must be added in indefinite integrals solutions. Remember the basic integration results.