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Chapter 4: Problem 12

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

Short Answer

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

Step by step solution

01

Simplify the Integral

First, take out the constant 3/2 from the integral to simplify it: \( \frac{3}{2} \int \frac{1}{\sqrt{x}(1+x)} d x \)
02

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: \( \frac{3}{2} \int \frac{1}{u} du \)
03

Integrate

Next, integrate the simplified expression. The integration of \( \frac{1}{u} \) is \( ln|u| \). Therefore, the integral becomes: \( \frac{3}{2} ln|u| + C \)
04

Back-substitution

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

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Most popular questions from this chapter

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