Problem 12

Question

Find the integral. $$ \int \frac{3}{2 \sqrt{x}(1+x)} d x $$

Step-by-Step Solution

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Answer

The integral of $ \frac{3}{2 \sqrt{x}(1+x)} d x $ is $ \frac{3}{2} ln|\sqrt{x}(1+x)| + C $

1Step 1: Simplify the Integral

First, take out the constant 3/2 from the integral to simplify it: $ \frac{3}{2} \int \frac{1}{\sqrt{x}(1+x)} d x $

2Step 2: Substitution

Now, let $ u = \sqrt{x}(1+x) $. Then, $ du = (\frac{1}{2\sqrt{x}} + \sqrt{x}) dx $. After substituting these into the integral, the integration becomes: $ \frac{3}{2} \int \frac{1}{u} du $

3Step 3: Integrate

Next, integrate the simplified expression. The integration of $ \frac{1}{u} $ is $ ln|u| $. Therefore, the integral becomes: $ \frac{3}{2} ln|u| + C $

4Step 4: Back-substitution

Lastly, substitute the original variable $ u = \sqrt{x}(1+x) $ back into the integral to get the final answer: $ \frac{3}{2} ln|\sqrt{x}(1+x)| + C $

Problem 12

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