Evaluate the integral:
$\int \frac{9}{49+x^2} d x$
The problem asks us to evaluate the indefinite integral \(\int \frac{9}{49+x^2} \, dx\). We recognize that the integrand is of a standard form involving the derivative of the inverse tangent function. Specifically, we can use the general rule \(\int \frac{1}{a^2+u^2} \, du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C\), where \(a\) is a constant. In this expression, \(a^2 = 49\), so \(a = 7\).
We can pull the constant factor \(9\) out of the integral:
\[\int \frac{9}{49+x^2} \, dx = 9 \int \frac{1}{7^2+x^2} \, dx\]
Now, we apply the standard integral formula with \(a = 7\):
\[9 \int \frac{1}{7^2+x^2} \, dx = 9 \cdot \left( \frac{1}{7} \arctan\left(\frac{x}{7}\right) \right) + C\]
Multiplying the constant \(9\) by the fraction \(\frac{1}{7}\), we get \(\frac{9}{7}\). Thus, the integral evaluates to:
\[\frac{9}{7} \arctan\left(\frac{x}{7}\right) + C\]
where \(C\) represents the constant of integration.
The final result of the indefinite integral \(\int \frac{9}{49+x^2} \, dx\) is \(\frac{9}{7} \arctan\left(\frac{x}{7}\right) + C\).
In physics and engineering, integrals of this form often appear when analyzing the electric field produced by a charged wire or when determining the time it takes for an object to reach a certain speed while experiencing air resistance proportional to the square of its velocity. For example, when calculating the velocity \(v(t)\) of an object in a medium where drag is proportional to \(v^2\), one might arrive at an expression involving the arctangent function, which is critical for predicting motion and design efficiency in aerodynamics.