Find the indefinite integral.
\[
\int \frac{x}{\sqrt{x-9}} d x
\]

Answer

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Answer

Final Answer: \(\boxed{\frac{2}{3}(x - 9)^{\frac{3}{2}} + C}\)

Steps

Step 1 ：Given the integral \(\int \frac{x}{\sqrt{x-9}} dx\)

Step 2 ：Let's use substitution to simplify it. Let \(u = x - 9\), then \(du = dx\). The integral becomes \(\int u^{\frac{1}{2}} du\)

Step 3 ：This is a standard form and can be integrated directly. The integral of \(u^{\frac{1}{2}}\) with respect to \(u\) is \(\frac{2}{3}u^{\frac{3}{2}}\)

Step 4 ：However, we need to convert it back to the original variable \(x\). So the final answer should be \(\frac{2}{3}(x - 9)^{\frac{3}{2}} + C\), where \(C\) is the constant of integration

Step 5 ：Final Answer: \(\boxed{\frac{2}{3}(x - 9)^{\frac{3}{2}} + C}\)

Find the indefinite integral.\[\int \frac{x}{\sqrt{x-9}} d x\]

Answer

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Find the indefinite integral.
\[
\int \frac{x}{\sqrt{x-9}} d x
\]