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integrate square root $\frac{y}{2-y}$

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Add and subtract three or more integers

Full solution

Q. integrate square root

\frac{y}{2-y}

Set Up Integral: Let's set up the integral that we need to solve. $\newline$ We need to integrate the function $\sqrt{\frac{y}{2-y}}$ with respect to $y$ . The integral is written as: $\newline$ $\int \sqrt{\frac{y}{2-y}} \, dy$
Perform Substitution: To solve this integral, we can perform a substitution. Let's let $u = 2 - y$ , which means that $du = -dy$ . We need to solve for $dy$ to substitute it back into our integral. Solving for $dy$ gives us $dy = -du$ .
Substitute $u$ and $dy$ : Substitute $u$ and $dy$ into the integral. $\newline$ Our integral now becomes: $\newline$ $\int\sqrt{\frac{y}{2-y}} (-du)$ $\newline$ Since $y = 2 - u$ , we can substitute $y$ as well to get: $\newline$ $\int\sqrt{\frac{2-u}{u}} (-du)$
Simplify Integral: Simplify the integral. $\newline$ The integral simplifies to: $\newline$ $-\int\sqrt{\left(\frac{2-u}{u}\right)} \, du$ $\newline$ This can be further simplified by splitting the square root into two parts: $\newline$ $-\int\sqrt{\frac{2}{u} - 1} \, du$
Complex Integral: This integral is not straightforward to solve. We may need to use a trigonometric substitution or partial fractions to solve it. However, this is a complex integral that may not have a simple antiderivative in terms of elementary functions. We will need to use advanced integration techniques that are beyond the scope of this solution.

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