Question 9 (4 points)
When evaluating the integral \( \int \frac{x^{3} d x}{\sqrt{4 x^{2}+36}} \) which of the following would be the best substitution for \( \boldsymbol{x} \) ?

Answer

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Answer

3. Simplify: \(\frac{1}{8} \int \frac{u^{3} du}{\sqrt{u^{2}+36}}\)

Steps

Step 1 ：1. Let \(u = 2x\) then \(\frac{du}{dx} = 2\)

Step 2 ：2. Replace x in the integral: \(\int \frac{\frac{u^3}{8} du}{\sqrt{u^{2}+36}}\)

Step 3 ：3. Simplify: \(\frac{1}{8} \int \frac{u^{3} du}{\sqrt{u^{2}+36}}\)