Have you ever wondered how to find the integral of \(\sqrt{x}\)? It’s actually quite simple once you know the power rule!
Let’s start by rewriting \(\sqrt{x}\) in a way that’s easier to work with. We know that:
\[ \sqrt{x} = x^{\frac{1}{2}} \]
Now, we can use the power rule for integration. The power rule says that to integrate \(x^n\), you add 1 to the exponent and then divide by the new exponent. So, let’s apply it:
\[ \int \sqrt{x} \, dx = \int x^{\frac{1}{2}} \, dx \]
Following the power rule:
\[ \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{3}{2}}}{\dfrac{3}{2}} + C = \frac{2}{3} x^{\frac{3}{2}} + C \]
This is our final answer:
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{\frac{3}{2}} + C \]
In simpler terms, we turned \(\sqrt{x}\) into a power of \(x\) (specifically, \(x^{\frac{1}{2}}\)), and then we applied the power rule. This same technique can help you integrate other square roots, like \(\sqrt{x^3}\). For example:
\[ \sqrt{x^3} = x^{\frac{3}{2}} \]
Now you can easily integrate functions that involve square roots by turning them into powers of \(x\).
