Evaluate the integral:
$\int \frac{x\, dx}{\sqrt{1+4x^2}}=$
The objective is to evaluate the indefinite integral \(\int \frac{x}{\sqrt{1+4x^2}} \, dx\). This is an integral of a rational function involving a square root, which is a classic candidate for the method of u-substitution. We aim to transform this integral into a simpler form that can be integrated using the basic power rule for integration.
To simplify the integrand, we select an appropriate substitution. Let \(u = 1 + 4x^2\). Next, we find the differential \(du\) by differentiating both sides with respect to \(x\):
\[\frac{du}{dx} = \frac{d}{dx}(1 + 4x^2) = 8x\]
This gives us \(du = 8x \, dx\). Looking at our original integral, we have the term \(x \, dx\), which can be written as \(x \, dx = \frac{1}{8} \, du\).
Now we rewrite the integral in terms of \(u\):
\[ \int \frac{x}{\sqrt{1+4x^2}} \, dx = \int \frac{1}{\sqrt{u}} \cdot \frac{1}{8} \, du = \frac{1}{8} \int u^{-1/2} \, du \]
We apply the power rule for integration, which states \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\) for \(n
eq -1\). Here, \(n = -1/2\), so:
\[ \int u^{-1/2} \, du = \frac{u^{(-1/2) + 1}}{(-1/2) + 1} + C = \frac{u^{1/2}}{1/2} + C = 2\sqrt{u} + C \]
Multiplying by the constant \(\frac{1}{8}\) gives:
\[ \frac{1}{8} \cdot 2\sqrt{u} + C = \frac{1}{4}\sqrt{u} + C \]
Finally, we substitute the original expression for \(u\) back into the result. Since \(u = 1 + 4x^2\), we have:
\[ \frac{1}{4}\sqrt{u} + C = \frac{1}{4}\sqrt{1+4x^2} + C \]
This is the antiderivative of the given function.
In physics, such integrals arise when calculating work done by a variable force or when determining the length of a curve in space. For instance, if a particle moves along a path defined by a function, the arc length integral often requires these substitution techniques to find the total distance traveled over a given interval, enabling precise tracking of physical systems.