Question

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

Short answer

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

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: \[\int \sqrt{x}\, dx + \int \frac{1}{2\sqrt{x}}\, dx\]

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 \].

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.