∫ cos ( t ) t d t {\displaystyle \int {\frac {\cos {({\sqrt {t}})}}{\sqrt {t}}}\;dt}
u = t d u = ( 1 2 1 t ) d t 2 d u = 1 t d t {\displaystyle {\begin{aligned}u&={\sqrt {t}}\\[2ex]du&=({\frac {1}{2}}\ {\frac {1}{\sqrt {t}}})\;dt\\[2ex]2du&={\frac {1}{\sqrt {t}}}\;dt\end{aligned}}}
∫ 1 t cos ( t ) d t = 2 ∫ cos u d u = 2 sin u + c = 2 sin ( t ) + c {\displaystyle {\begin{aligned}\int {\frac {1}{\sqrt {t}}}\cos {({\sqrt {t}})}dt&=2\int \cos {u}\;du\\[2ex]&=2\sin {u}+c\\[2ex]&=2\sin({\sqrt {t}})+c\\[2ex]\end{aligned}}}
![{\displaystyle {\begin{aligned}u&={\sqrt {t}}\\[2ex]du&=({\frac {1}{2}}\ {\frac {1}{\sqrt {t}}})\;dt\\[2ex]2du&={\frac {1}{\sqrt {t}}}\;dt\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0ac506fb861c0050d06ca2164528e03909ec6b6e)
![{\displaystyle {\begin{aligned}\int {\frac {1}{\sqrt {t}}}\cos {({\sqrt {t}})}dt&=2\int \cos {u}\;du\\[2ex]&=2\sin {u}+c\\[2ex]&=2\sin({\sqrt {t}})+c\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cab9134887b680d923404d9383d7bab4ad18629e)