$\displaystyle \int \cos\sqrt{7t} dt$

$\displaystyle \int \cos(7t)^{1/2} dt$

$\displaystyle \int \cos (u)^{1/2} dt$

$\displaystyle u = 7t$

$\displaystyle du = 7$

$\displaystyle \dfrac{1}{7}du = dt$

$\displaystyle \dfrac{1}{7}\int \cos u^{1/2}$

$\displaystyle \dfrac{1}{7}\int \dfrac{\sin u^{3/2}}{\dfrac{3}{2}} + C$

$\displaystyle (\dfrac{1}{7})(\dfrac{2}{3})\int \sin u^{3/2} + C$

$\displaystyle (\dfrac{1}{7})(\dfrac{2}{3})\int \sin u^{3/2} + C$

$\displaystyle \dfrac{2}{21}\int \sin u^{3/2} + C$

$\displaystyle \dfrac{2}{21}\int \sin 7t^{3/2} + C$