175. $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$.

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175. $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$.
Solution. $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x=\left|\begin{array}{l}\sqrt{x}=t \\ \frac{d x}{2 \sqrt{x}}=d t \\ \frac{d x}{\sqrt{x}}=2 d t\end{array}\right|=2 \int \sin t d t=-2 \cos t+C=$ $=-2 \cos \sqrt{x}+C$.

This integral can also be found by bringing $\sqrt{x}$ under the differential sign:

$$
\int \frac{\sin \sqrt{x}}{\sqrt{x}} d^{2} x=2 \int \sin \sqrt{x} d(\sqrt{x})=-2 \cos \sqrt{x}+C
$$