An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the cumulative sum of infinitely small values over a given interval. It is the inverse operation of differentiation.
To solve an integral, you need to find the antiderivative of the given function. This can be done by using integration techniques such as substitution, integration by parts, or using a table of integrals. Once you have found the antiderivative, you can evaluate the integral at the given limits to find the final answer.
The given integral is asking for the antiderivative of the function \frac{x+9}{x^2+9}. This means that we need to find a function whose derivative is equal to \frac{x+9}{x^2+9}.
To solve \int \frac{x+9}{x^2+9} dx, we can first use the substitution u = x^2+9. This will give us du = 2x dx, which we can use to replace x dx in the original integral. This will result in \frac{1}{2}\int \frac{1}{u} du. We can then use the rule \int \frac{1}{u} du = \ln u + C to find the antiderivative. Finally, we can substitute back u = x^2+9 to get the final answer of \frac{1}{2}\ln(x^2+9) + C.
Yes, there are some restrictions for solving this integral. The function \frac{x+9}{x^2+9} is undefined at x = \pm 3 because it results in a division by zero. Therefore, the limits of integration must not include these values. Additionally, the substitution u = x^2+9 only works for x \neq \pm 3. If the limits of integration do include these values, a different substitution or integration technique must be used.