Substitution with u = sqrt(x)

Let $u = \sqrt{x}$. Then $du = \frac{1}{2\sqrt{x}}\,dx$, so $2\,du = \frac{dx}{\sqrt{x}}$. Substituting into the integral gives $\int \cos(u)\cdot 2\,du = 2\int \cos(u)\,du$.

Evaluating this simpler integral yields $2\sin(u) + C$. Substituting back $u = \sqrt{x}$ gives the final answer $2\sin(\sqrt{x}) + C$.

The answer is two. Since u equals the square root of x, its derivative introduces a factor of one half in du, which means dx over the square root of x equals two du. That constant two carries through as the coefficient multiplying cosine of u inside the integral before integrating.