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** Step-by-step Solution **

Problem to solve:

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Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

Learn how to solve integrals with radicals problems step by step online.

$\int x^{-\frac{1}{2}}\sin\left(\sqrt{x}\right)dx$

Learn how to solve integrals with radicals problems step by step online. Integrate int((sin(x^1/2)/(x^1/2))dx. Rewrite the exponent using the power rule \frac{a^m}{a^n}=a^{m-n}, where in this case m=0. We can solve the integral \int x^{-\frac{1}{2}}\sin\left(\sqrt{x}\right)dx by applying integration by parts method to calculate the integral of the product of two functions, using the following formula. First, identify u and calculate du. Now, identify dv and calculate v.

** Final Answer

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