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Chapter 11: Problem 32

Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x $$

Short Answer

Expert verified
The indefinite integral of the given function is \(\frac{2}{3}x^\frac{3}{2} + 2x^\frac{1}{2} + C \), and it is confirmed by differentiation that the result is correct.

Step by step solution

01

Breaking down the overall integral into individual integrals

The integral can be separated into two parts using the rule of the integral of the sum. This gives solutions: \[\int \sqrt{x}\, dx + \int \frac{1}{2\sqrt{x}}\, dx\]
02

Integration

Now each term can be integrated using the power rule of integration. For a power of n (where n is not -1), the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). \n This results in \[\frac{x^\frac{3}{2}}{\frac{3}{2}} + \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C\], where C is an arbitrary constant. Simplifying, it results to the equation: \(\frac{2}{3}x^\frac{3}{2} + 2x^\frac{1}{2} + C \].
03

Confirm by differentiation

To confirm this solution, differentiate the result using the power rule for differentiation \[f'(x) = n x^{n-1}\]. This results in \[\frac{2}{3}\cdot \frac{3}{2} x^{\frac{3}{2}-1} + 2\cdot \frac{1}{2} x^{\frac{1}{2}-1}\], or, by simplifying, \[\sqrt{x}+ \frac{1}{2\sqrt{x}}\], thus confirming the correctness of the original calculation.

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